
huffman@21164: (*  Title       : Deriv.thy
huffman@21164:     ID          : $Id$
huffman@21164:     Author      : Jacques D. Fleuriot
huffman@21164:     Copyright   : 1998  University of Cambridge
huffman@21164:     Conversion to Isar and new proofs by Lawrence C Paulson, 2004
huffman@21164:     GMVT by Benjamin Porter, 2005
huffman@21164: *)
huffman@21164: 
huffman@21164: header{* Differentiation *}
huffman@21164: 
huffman@21164: theory Deriv
huffman@21164: imports Lim
huffman@21164: begin
huffman@21164: 
huffman@21164: text{*Standard and Nonstandard Definitions*}
huffman@21164: 
huffman@21164: definition
huffman@21164:   deriv :: "[real \<Rightarrow> 'a::real_normed_vector, real, 'a] \<Rightarrow> bool"
huffman@21164:     --{*Differentiation: D is derivative of function f at x*}
wenzelm@21404:           ("(DERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60) where
huffman@21164:   "DERIV f x :> D = ((%h. (f(x + h) - f x) /# h) -- 0 --> D)"
huffman@21164: 
wenzelm@21404: definition
huffman@21164:   nsderiv :: "[real=>real,real,real] => bool"
wenzelm@21404:           ("(NSDERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60) where
huffman@21164:   "NSDERIV f x :> D = (\<forall>h \<in> Infinitesimal - {0}.
huffman@21164:       (( *f* f)(hypreal_of_real x + h)
huffman@21164:        - hypreal_of_real (f x))/h @= hypreal_of_real D)"
huffman@21164: 
wenzelm@21404: definition
wenzelm@21404:   differentiable :: "[real=>real,real] => bool"   (infixl "differentiable" 60) where
huffman@21164:   "f differentiable x = (\<exists>D. DERIV f x :> D)"
huffman@21164: 
wenzelm@21404: definition
huffman@21164:   NSdifferentiable :: "[real=>real,real] => bool"
wenzelm@21404:                        (infixl "NSdifferentiable" 60) where
huffman@21164:   "f NSdifferentiable x = (\<exists>D. NSDERIV f x :> D)"
huffman@21164: 
wenzelm@21404: definition
wenzelm@21404:   increment :: "[real=>real,real,hypreal] => hypreal" where
huffman@21164:   "increment f x h = (@inc. f NSdifferentiable x &
huffman@21164:            inc = ( *f* f)(hypreal_of_real x + h) - hypreal_of_real (f x))"
huffman@21164: 
huffman@21164: 
huffman@21164: consts
huffman@21164:   Bolzano_bisect :: "[real*real=>bool, real, real, nat] => (real*real)"
huffman@21164: primrec
huffman@21164:   "Bolzano_bisect P a b 0 = (a,b)"
huffman@21164:   "Bolzano_bisect P a b (Suc n) =
huffman@21164:       (let (x,y) = Bolzano_bisect P a b n
huffman@21164:        in if P(x, (x+y)/2) then ((x+y)/2, y)
huffman@21164:                             else (x, (x+y)/2))"
huffman@21164: 
huffman@21164: 
huffman@21164: subsection {* Derivatives *}
huffman@21164: 
huffman@21164: subsubsection {* Purely standard proofs *}
huffman@21164: 
huffman@21164: lemma DERIV_iff: "(DERIV f x :> D) = ((%h. (f(x + h) - f(x))/#h) -- 0 --> D)"
huffman@21164: by (simp add: deriv_def)
huffman@21164: 
huffman@21164: lemma DERIV_D: "DERIV f x :> D ==> (%h. (f(x + h) - f(x))/#h) -- 0 --> D"
huffman@21164: by (simp add: deriv_def)
huffman@21164: 
huffman@21164: lemma DERIV_const [simp]: "DERIV (\<lambda>x. k) x :> 0"
huffman@21164: by (simp add: deriv_def)
huffman@21164: 
huffman@21164: lemma DERIV_Id [simp]: "DERIV (\<lambda>x. x) x :> 1"
huffman@21164: by (simp add: deriv_def real_scaleR_def cong: LIM_cong)
huffman@21164: 
huffman@21164: lemma add_diff_add:
huffman@21164:   fixes a b c d :: "'a::ab_group_add"
huffman@21164:   shows "(a + c) - (b + d) = (a - b) + (c - d)"
huffman@21164: by simp
huffman@21164: 
huffman@21164: lemma DERIV_add:
huffman@21164:   "\<lbrakk>DERIV f x :> D; DERIV g x :> E\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. f x + g x) x :> D + E"
huffman@21164: by (simp only: deriv_def add_diff_add scaleR_right_distrib LIM_add)
huffman@21164: 
huffman@21164: lemma DERIV_minus:
huffman@21164:   "DERIV f x :> D \<Longrightarrow> DERIV (\<lambda>x. - f x) x :> - D"
huffman@21164: by (simp only: deriv_def minus_diff_minus scaleR_minus_right LIM_minus)
huffman@21164: 
huffman@21164: lemma DERIV_diff:
huffman@21164:   "\<lbrakk>DERIV f x :> D; DERIV g x :> E\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. f x - g x) x :> D - E"
huffman@21164: by (simp only: diff_def DERIV_add DERIV_minus)
huffman@21164: 
huffman@21164: lemma DERIV_add_minus:
huffman@21164:   "\<lbrakk>DERIV f x :> D; DERIV g x :> E\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. f x + - g x) x :> D + - E"
huffman@21164: by (simp only: DERIV_add DERIV_minus)
huffman@21164: 
huffman@21164: lemma DERIV_isCont: "DERIV f x :> D \<Longrightarrow> isCont f x"
huffman@21164: proof (unfold isCont_iff)
huffman@21164:   assume "DERIV f x :> D"
huffman@21164:   hence "(\<lambda>h. (f(x+h) - f(x)) /# h) -- 0 --> D"
huffman@21164:     by (rule DERIV_D)
huffman@21164:   hence "(\<lambda>h. h *# ((f(x+h) - f(x)) /# h)) -- 0 --> 0 *# D"
huffman@21164:     by (intro LIM_scaleR LIM_self)
huffman@21164:   hence "(\<lambda>h. (f(x+h) - f(x))) -- 0 --> 0"
huffman@21164:     by (simp cong: LIM_cong)
huffman@21164:   thus "(\<lambda>h. f(x+h)) -- 0 --> f(x)"
huffman@21164:     by (simp add: LIM_def)
huffman@21164: qed
huffman@21164: 
huffman@21164: lemma DERIV_mult_lemma:
huffman@21164:   fixes a b c d :: "'a::real_algebra"
huffman@21164:   shows "(a * b - c * d) /# h = a * ((b - d) /# h) + ((a - c) /# h) * d"
huffman@21164: by (simp add: diff_minus scaleR_right_distrib [symmetric] ring_distrib)
huffman@21164: 
huffman@21164: lemma DERIV_mult':
huffman@21164:   fixes f g :: "real \<Rightarrow> 'a::real_normed_algebra"
huffman@21164:   assumes f: "DERIV f x :> D"
huffman@21164:   assumes g: "DERIV g x :> E"
huffman@21164:   shows "DERIV (\<lambda>x. f x * g x) x :> f x * E + D * g x"
huffman@21164: proof (unfold deriv_def)
huffman@21164:   from f have "isCont f x"
huffman@21164:     by (rule DERIV_isCont)
huffman@21164:   hence "(\<lambda>h. f(x+h)) -- 0 --> f x"
huffman@21164:     by (simp only: isCont_iff)
huffman@21164:   hence "(\<lambda>h. f(x+h) * ((g(x+h) - g x) /# h) +
huffman@21164:               ((f(x+h) - f x) /# h) * g x)
huffman@21164:           -- 0 --> f x * E + D * g x"
huffman@21164:     by (intro LIM_add LIM_mult2 LIM_const DERIV_D f g)
huffman@21164:   thus "(\<lambda>h. (f(x+h) * g(x+h) - f x * g x) /# h)
huffman@21164:          -- 0 --> f x * E + D * g x"
huffman@21164:     by (simp only: DERIV_mult_lemma)
huffman@21164: qed
huffman@21164: 
huffman@21164: lemma DERIV_mult:
huffman@21164:   fixes f g :: "real \<Rightarrow> 'a::{real_normed_algebra,comm_ring}" shows
huffman@21164:      "[| DERIV f x :> Da; DERIV g x :> Db |]
huffman@21164:       ==> DERIV (%x. f x * g x) x :> (Da * g(x)) + (Db * f(x))"
huffman@21164: by (drule (1) DERIV_mult', simp only: mult_commute add_commute)
huffman@21164: 
huffman@21164: lemma DERIV_unique:
huffman@21164:       "[| DERIV f x :> D; DERIV f x :> E |] ==> D = E"
huffman@21164: apply (simp add: deriv_def)
huffman@21164: apply (blast intro: LIM_unique)
huffman@21164: done
huffman@21164: 
huffman@21164: text{*Differentiation of finite sum*}
huffman@21164: 
huffman@21164: lemma DERIV_sumr [rule_format (no_asm)]:
huffman@21164:      "(\<forall>r. m \<le> r & r < (m + n) --> DERIV (%x. f r x) x :> (f' r x))
huffman@21164:       --> DERIV (%x. \<Sum>n=m..<n::nat. f n x :: real) x :> (\<Sum>r=m..<n. f' r x)"
huffman@21164: apply (induct "n")
huffman@21164: apply (auto intro: DERIV_add)
huffman@21164: done
huffman@21164: 
huffman@21164: text{*Alternative definition for differentiability*}
huffman@21164: 
huffman@21164: lemma DERIV_LIM_iff:
huffman@21164:      "((%h::real. (f(a + h) - f(a)) / h) -- 0 --> D) =
huffman@21164:       ((%x. (f(x)-f(a)) / (x-a)) -- a --> D)"
huffman@21164: apply (rule iffI)
huffman@21164: apply (drule_tac k="- a" in LIM_offset)
huffman@21164: apply (simp add: diff_minus)
huffman@21164: apply (drule_tac k="a" in LIM_offset)
huffman@21164: apply (simp add: add_commute)
huffman@21164: done
huffman@21164: 
huffman@21164: lemma DERIV_LIM_iff':
huffman@21164:      "((%h::real. (f(a + h) - f(a)) /# h) -- 0 --> D) =
huffman@21164:       ((%x. (f(x)-f(a)) /# (x-a)) -- a --> D)"
huffman@21164: apply (rule iffI)
huffman@21164: apply (drule_tac k="- a" in LIM_offset)
huffman@21164: apply (simp add: diff_minus)
huffman@21164: apply (drule_tac k="a" in LIM_offset)
huffman@21164: apply (simp add: add_commute)
huffman@21164: done
huffman@21164: 
huffman@21164: lemma DERIV_iff2: "(DERIV f x :> D) = ((%z. (f(z) - f(x)) /# (z-x)) -- x --> D)"
huffman@21164: by (simp add: deriv_def diff_minus [symmetric] DERIV_LIM_iff')
huffman@21164: 
huffman@21164: lemma inverse_diff_inverse:
huffman@21164:   "\<lbrakk>(a::'a::division_ring) \<noteq> 0; b \<noteq> 0\<rbrakk>
huffman@21164:    \<Longrightarrow> inverse a - inverse b = - (inverse a * (a - b) * inverse b)"
huffman@21164: by (simp add: right_diff_distrib left_diff_distrib mult_assoc)
huffman@21164: 
huffman@21164: lemma DERIV_inverse_lemma:
huffman@21164:   "\<lbrakk>a \<noteq> 0; b \<noteq> (0::'a::real_normed_div_algebra)\<rbrakk>
huffman@21164:    \<Longrightarrow> (inverse a - inverse b) /# h
huffman@21164:      = - (inverse a * ((a - b) /# h) * inverse b)"
huffman@21164: by (simp add: inverse_diff_inverse)
huffman@21164: 
huffman@21164: lemma LIM_equal2:
huffman@21164:   assumes 1: "0 < R"
huffman@21164:   assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < R\<rbrakk> \<Longrightarrow> f x = g x"
huffman@21164:   shows "g -- a --> l \<Longrightarrow> f -- a --> l"
huffman@21164: apply (unfold LIM_def, safe)
huffman@21164: apply (drule_tac x="r" in spec, safe)
huffman@21164: apply (rule_tac x="min s R" in exI, safe)
huffman@21164: apply (simp add: 1)
huffman@21164: apply (simp add: 2)
huffman@21164: done
huffman@21164: 
huffman@21164: lemma DERIV_inverse':
huffman@21164:   fixes f :: "real \<Rightarrow> 'a::real_normed_div_algebra"
huffman@21164:   assumes der: "DERIV f x :> D"
huffman@21164:   assumes neq: "f x \<noteq> 0"
huffman@21164:   shows "DERIV (\<lambda>x. inverse (f x)) x :> - (inverse (f x) * D * inverse (f x))"
huffman@21164:     (is "DERIV _ _ :> ?E")
huffman@21164: proof (unfold DERIV_iff2)
huffman@21164:   from der have lim_f: "f -- x --> f x"
huffman@21164:     by (rule DERIV_isCont [unfolded isCont_def])
huffman@21164: 
huffman@21164:   from neq have "0 < norm (f x)" by simp
huffman@21164:   with LIM_D [OF lim_f] obtain s
huffman@21164:     where s: "0 < s"
huffman@21164:     and less_fx: "\<And>z. \<lbrakk>z \<noteq> x; norm (z - x) < s\<rbrakk>
huffman@21164:                   \<Longrightarrow> norm (f z - f x) < norm (f x)"
huffman@21164:     by fast
huffman@21164: 
huffman@21164:   show "(\<lambda>z. (inverse (f z) - inverse (f x)) /# (z - x)) -- x --> ?E"
huffman@21164:   proof (rule LIM_equal2 [OF s])
huffman@21164:     fix z :: real
huffman@21164:     assume "z \<noteq> x" "norm (z - x) < s"
huffman@21164:     hence "norm (f z - f x) < norm (f x)" by (rule less_fx)
huffman@21164:     hence "f z \<noteq> 0" by auto
huffman@21164:     thus "(inverse (f z) - inverse (f x)) /# (z - x) =
huffman@21164:           - (inverse (f z) * ((f z - f x) /# (z - x)) * inverse (f x))"
huffman@21164:       using neq by (rule DERIV_inverse_lemma)
huffman@21164:   next
huffman@21164:     from der have "(\<lambda>z. (f z - f x) /# (z - x)) -- x --> D"
huffman@21164:       by (unfold DERIV_iff2)
huffman@21164:     thus "(\<lambda>z. - (inverse (f z) * ((f z - f x) /# (z - x)) * inverse (f x)))
huffman@21164:           -- x --> ?E"
huffman@21164:       by (intro LIM_mult2 LIM_inverse LIM_minus LIM_const lim_f neq)
huffman@21164:   qed
huffman@21164: qed
huffman@21164: 
huffman@21164: lemma DERIV_divide:
huffman@21164:   fixes D E :: "'a::{real_normed_div_algebra,field}"
huffman@21164:   shows "\<lbrakk>DERIV f x :> D; DERIV g x :> E; g x \<noteq> 0\<rbrakk>
huffman@21164:          \<Longrightarrow> DERIV (\<lambda>x. f x / g x) x :> (D * g x - f x * E) / (g x * g x)"
huffman@21164: apply (subgoal_tac "f x * - (inverse (g x) * E * inverse (g x)) +
huffman@21164:           D * inverse (g x) = (D * g x - f x * E) / (g x * g x)")
huffman@21164: apply (erule subst)
huffman@21164: apply (unfold divide_inverse)
huffman@21164: apply (erule DERIV_mult')
huffman@21164: apply (erule (1) DERIV_inverse')
huffman@21164: apply (simp add: left_diff_distrib nonzero_inverse_mult_distrib)
huffman@21164: apply (simp add: mult_ac)
huffman@21164: done
huffman@21164: 
huffman@21164: lemma DERIV_power_Suc:
krauss@21199:   fixes f :: "real \<Rightarrow> 'a::{real_normed_algebra,recpower,comm_monoid_mult}"
huffman@21164:   assumes f: "DERIV f x :> D"
huffman@21164:   shows "DERIV (\<lambda>x. f x ^ Suc n) x :> (of_nat n + 1) *# (D * f x ^ n)"
huffman@21164: proof (induct n)
huffman@21164: case 0
huffman@21164:   show ?case by (simp add: power_Suc f)
huffman@21164: case (Suc k)
huffman@21164:   from DERIV_mult' [OF f Suc] show ?case
huffman@21164:     apply (simp only: of_nat_Suc scaleR_left_distrib scaleR_one)
huffman@21164:     apply (simp only: power_Suc right_distrib mult_scaleR_right mult_ac)
huffman@21164:     done
huffman@21164: qed
huffman@21164: 
huffman@21164: lemma DERIV_power:
krauss@21199:   fixes f :: "real \<Rightarrow> 'a::{real_normed_algebra,recpower,comm_monoid_mult}"
huffman@21164:   assumes f: "DERIV f x :> D"
huffman@21164:   shows "DERIV (\<lambda>x. f x ^ n) x :> of_nat n *# (D * f x ^ (n - Suc 0))"
huffman@21164: by (cases "n", simp, simp add: DERIV_power_Suc f)
huffman@21164: 
huffman@21164: 
huffman@21164: (* ------------------------------------------------------------------------ *)
huffman@21164: (* Caratheodory formulation of derivative at a point: standard proof        *)
huffman@21164: (* ------------------------------------------------------------------------ *)
huffman@21164: 
huffman@21164: lemma CARAT_DERIV:
huffman@21164:      "(DERIV f x :> l) =
huffman@21164:       (\<exists>g. (\<forall>z. f z - f x = (z-x) *# g z) & isCont g x & g x = l)"
huffman@21164:       (is "?lhs = ?rhs")
huffman@21164: proof
huffman@21164:   assume der: "DERIV f x :> l"
huffman@21164:   show "\<exists>g. (\<forall>z. f z - f x = (z-x) *# g z) \<and> isCont g x \<and> g x = l"
huffman@21164:   proof (intro exI conjI)
huffman@21164:     let ?g = "(%z. if z = x then l else (f z - f x) /# (z-x))"
huffman@21164:     show "\<forall>z. f z - f x = (z-x) *# ?g z" by (simp)
huffman@21164:     show "isCont ?g x" using der
huffman@21164:       by (simp add: isCont_iff DERIV_iff diff_minus
huffman@21164:                cong: LIM_equal [rule_format])
huffman@21164:     show "?g x = l" by simp
huffman@21164:   qed
huffman@21164: next
huffman@21164:   assume "?rhs"
huffman@21164:   then obtain g where
huffman@21164:     "(\<forall>z. f z - f x = (z-x) *# g z)" and "isCont g x" and "g x = l" by blast
huffman@21164:   thus "(DERIV f x :> l)"
huffman@21164:      by (auto simp add: isCont_iff DERIV_iff diff_minus
huffman@21164:                cong: LIM_equal [rule_format])
huffman@21164: qed
huffman@21164: 
huffman@21164: lemma DERIV_chain':
huffman@21164:   assumes f: "DERIV f x :> D"
huffman@21164:   assumes g: "DERIV g (f x) :> E"
huffman@21164:   shows "DERIV (\<lambda>x. g (f x)) x :> D *# E"
huffman@21164: proof (unfold DERIV_iff2)
huffman@21164:   obtain d where d: "\<forall>y. g y - g (f x) = (y - f x) *# d y"
huffman@21164:     and cont_d: "isCont d (f x)" and dfx: "d (f x) = E"
huffman@21164:     using CARAT_DERIV [THEN iffD1, OF g] by fast
huffman@21164:   from f have "f -- x --> f x"
huffman@21164:     by (rule DERIV_isCont [unfolded isCont_def])
huffman@21164:   with cont_d have "(\<lambda>z. d (f z)) -- x --> d (f x)"
huffman@21239:     by (rule isCont_LIM_compose)
huffman@21164:   hence "(\<lambda>z. ((f z - f x) /# (z - x)) *# d (f z))
huffman@21164:           -- x --> D *# d (f x)"
huffman@21164:     by (rule LIM_scaleR [OF f [unfolded DERIV_iff2]])
huffman@21164:   thus "(\<lambda>z. (g (f z) - g (f x)) /# (z - x)) -- x --> D *# E"
huffman@21164:     by (simp add: d dfx real_scaleR_def)
huffman@21164: qed
huffman@21164: 
huffman@21164: 
huffman@21164: subsubsection {* Nonstandard proofs *}
huffman@21164: 
huffman@21164: lemma DERIV_NS_iff:
huffman@21164:       "(DERIV f x :> D) = ((%h. (f(x + h) - f(x))/#h) -- 0 --NS> D)"
huffman@21164: by (simp add: deriv_def LIM_NSLIM_iff)
huffman@21164: 
huffman@21164: lemma NS_DERIV_D: "DERIV f x :> D ==> (%h. (f(x + h) - f(x))/#h) -- 0 --NS> D"
huffman@21164: by (simp add: deriv_def LIM_NSLIM_iff)
huffman@21164: 
huffman@21164: lemma NSDeriv_unique:
huffman@21164:      "[| NSDERIV f x :> D; NSDERIV f x :> E |] ==> D = E"
huffman@21164: apply (simp add: nsderiv_def)
huffman@21164: apply (cut_tac Infinitesimal_epsilon hypreal_epsilon_not_zero)
huffman@21164: apply (auto dest!: bspec [where x=epsilon]
huffman@21164:             intro!: inj_hypreal_of_real [THEN injD]
huffman@21164:             dest: approx_trans3)
huffman@21164: done
huffman@21164: 
huffman@21164: text {*First NSDERIV in terms of NSLIM*}
huffman@21164: 
huffman@21164: text{*first equivalence *}
huffman@21164: lemma NSDERIV_NSLIM_iff:
huffman@21164:       "(NSDERIV f x :> D) = ((%h. (f(x + h) - f(x))/h) -- 0 --NS> D)"
huffman@21164: apply (simp add: nsderiv_def NSLIM_def, auto)
huffman@21164: apply (drule_tac x = xa in bspec)
huffman@21164: apply (rule_tac [3] ccontr)
huffman@21164: apply (drule_tac [3] x = h in spec)
huffman@21164: apply (auto simp add: mem_infmal_iff starfun_lambda_cancel)
huffman@21164: done
huffman@21164: 
huffman@21164: text{*second equivalence *}
huffman@21164: lemma NSDERIV_NSLIM_iff2:
huffman@21164:      "(NSDERIV f x :> D) = ((%z. (f(z) - f(x)) / (z-x)) -- x --NS> D)"
huffman@21164: by (simp add: NSDERIV_NSLIM_iff DERIV_LIM_iff  diff_minus [symmetric]
huffman@21164:               LIM_NSLIM_iff [symmetric])
huffman@21164: 
huffman@21164: (* while we're at it! *)
huffman@21164: lemma NSDERIV_iff2:
huffman@21164:      "(NSDERIV f x :> D) =
huffman@21164:       (\<forall>w.
huffman@21164:         w \<noteq> hypreal_of_real x & w \<approx> hypreal_of_real x -->
huffman@21164:         ( *f* (%z. (f z - f x) / (z-x))) w \<approx> hypreal_of_real D)"
huffman@21164: by (simp add: NSDERIV_NSLIM_iff2 NSLIM_def)
huffman@21164: 
huffman@21164: (*FIXME DELETE*)
huffman@21164: lemma hypreal_not_eq_minus_iff: "(x \<noteq> a) = (x - a \<noteq> (0::hypreal))"
huffman@21164: by (auto dest: hypreal_eq_minus_iff [THEN iffD2])
huffman@21164: 
huffman@21164: lemma NSDERIVD5:
huffman@21164:   "(NSDERIV f x :> D) ==>
huffman@21164:    (\<forall>u. u \<approx> hypreal_of_real x -->
huffman@21164:      ( *f* (%z. f z - f x)) u \<approx> hypreal_of_real D * (u - hypreal_of_real x))"
huffman@21164: apply (auto simp add: NSDERIV_iff2)
huffman@21164: apply (case_tac "u = hypreal_of_real x", auto)
huffman@21164: apply (drule_tac x = u in spec, auto)
huffman@21164: apply (drule_tac c = "u - hypreal_of_real x" and b = "hypreal_of_real D" in approx_mult1)
huffman@21164: apply (drule_tac [!] hypreal_not_eq_minus_iff [THEN iffD1])
huffman@21164: apply (subgoal_tac [2] "( *f* (%z. z-x)) u \<noteq> (0::hypreal) ")
huffman@21164: apply (auto simp add:
huffman@21164:          approx_minus_iff [THEN iffD1, THEN mem_infmal_iff [THEN iffD2]]
huffman@21164:          Infinitesimal_subset_HFinite [THEN subsetD])
huffman@21164: done
huffman@21164: 
huffman@21164: lemma NSDERIVD4:
huffman@21164:      "(NSDERIV f x :> D) ==>
huffman@21164:       (\<forall>h \<in> Infinitesimal.
huffman@21164:                (( *f* f)(hypreal_of_real x + h) -
huffman@21164:                  hypreal_of_real (f x))\<approx> (hypreal_of_real D) * h)"
huffman@21164: apply (auto simp add: nsderiv_def)
huffman@21164: apply (case_tac "h = (0::hypreal) ")
huffman@21164: apply (auto simp add: diff_minus)
huffman@21164: apply (drule_tac x = h in bspec)
huffman@21164: apply (drule_tac [2] c = h in approx_mult1)
huffman@21164: apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
huffman@21164:             simp add: diff_minus)
huffman@21164: done
huffman@21164: 
huffman@21164: lemma NSDERIVD3:
huffman@21164:      "(NSDERIV f x :> D) ==>
huffman@21164:       (\<forall>h \<in> Infinitesimal - {0}.
huffman@21164:                (( *f* f)(hypreal_of_real x + h) -
huffman@21164:                  hypreal_of_real (f x))\<approx> (hypreal_of_real D) * h)"
huffman@21164: apply (auto simp add: nsderiv_def)
huffman@21164: apply (rule ccontr, drule_tac x = h in bspec)
huffman@21164: apply (drule_tac [2] c = h in approx_mult1)
huffman@21164: apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
huffman@21164:             simp add: mult_assoc diff_minus)
huffman@21164: done
huffman@21164: 
huffman@21164: text{*Differentiability implies continuity
huffman@21164:          nice and simple "algebraic" proof*}
huffman@21164: lemma NSDERIV_isNSCont: "NSDERIV f x :> D ==> isNSCont f x"
huffman@21164: apply (auto simp add: nsderiv_def isNSCont_NSLIM_iff NSLIM_def)
huffman@21164: apply (drule approx_minus_iff [THEN iffD1])
huffman@21164: apply (drule hypreal_not_eq_minus_iff [THEN iffD1])
huffman@21164: apply (drule_tac x = "xa - hypreal_of_real x" in bspec)
huffman@21164:  prefer 2 apply (simp add: add_assoc [symmetric])
huffman@21164: apply (auto simp add: mem_infmal_iff [symmetric] add_commute)
huffman@21164: apply (drule_tac c = "xa - hypreal_of_real x" in approx_mult1)
huffman@21164: apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
huffman@21164:             simp add: mult_assoc)
huffman@21164: apply (drule_tac x3=D in
huffman@21164:            HFinite_hypreal_of_real [THEN [2] Infinitesimal_HFinite_mult,
huffman@21164:              THEN mem_infmal_iff [THEN iffD1]])
huffman@21164: apply (auto simp add: mult_commute
huffman@21164:             intro: approx_trans approx_minus_iff [THEN iffD2])
huffman@21164: done
huffman@21164: 
huffman@21164: text{*Differentiation rules for combinations of functions
huffman@21164:       follow from clear, straightforard, algebraic
huffman@21164:       manipulations*}
huffman@21164: text{*Constant function*}
huffman@21164: 
huffman@21164: (* use simple constant nslimit theorem *)
huffman@21164: lemma NSDERIV_const [simp]: "(NSDERIV (%x. k) x :> 0)"
huffman@21164: by (simp add: NSDERIV_NSLIM_iff)
huffman@21164: 
huffman@21164: text{*Sum of functions- proved easily*}
huffman@21164: 
huffman@21164: lemma NSDERIV_add: "[| NSDERIV f x :> Da;  NSDERIV g x :> Db |]
huffman@21164:       ==> NSDERIV (%x. f x + g x) x :> Da + Db"
huffman@21164: apply (auto simp add: NSDERIV_NSLIM_iff NSLIM_def)
huffman@21164: apply (auto simp add: add_divide_distrib diff_divide_distrib dest!: spec)
huffman@21164: apply (drule_tac b = "hypreal_of_real Da" and d = "hypreal_of_real Db" in approx_add)
huffman@21164: apply (auto simp add: diff_def add_ac)
huffman@21164: done
huffman@21164: 
huffman@21164: text{*Product of functions - Proof is trivial but tedious
huffman@21164:   and long due to rearrangement of terms*}
huffman@21164: 
huffman@21164: lemma lemma_nsderiv1: "((a::hypreal)*b) - (c*d) = (b*(a - c)) + (c*(b - d))"
huffman@21164: by (simp add: right_diff_distrib)
huffman@21164: 
huffman@21164: lemma lemma_nsderiv2: "[| (x - y) / z = hypreal_of_real D + yb; z \<noteq> 0;
huffman@21164:          z \<in> Infinitesimal; yb \<in> Infinitesimal |]
huffman@21164:       ==> x - y \<approx> 0"
huffman@21164: apply (frule_tac c1 = z in hypreal_mult_right_cancel [THEN iffD2], assumption)
huffman@21164: apply (erule_tac V = "(x - y) / z = hypreal_of_real D + yb" in thin_rl)
huffman@21164: apply (auto intro!: Infinitesimal_HFinite_mult2 HFinite_add
huffman@21164:             simp add: mult_assoc mem_infmal_iff [symmetric])
huffman@21164: apply (erule Infinitesimal_subset_HFinite [THEN subsetD])
huffman@21164: done
huffman@21164: 
huffman@21164: lemma NSDERIV_mult: "[| NSDERIV f x :> Da; NSDERIV g x :> Db |]
huffman@21164:       ==> NSDERIV (%x. f x * g x) x :> (Da * g(x)) + (Db * f(x))"
huffman@21164: apply (auto simp add: NSDERIV_NSLIM_iff NSLIM_def)
huffman@21164: apply (auto dest!: spec
huffman@21164:       simp add: starfun_lambda_cancel lemma_nsderiv1)
huffman@21164: apply (simp (no_asm) add: add_divide_distrib diff_divide_distrib)
huffman@21164: apply (drule bex_Infinitesimal_iff2 [THEN iffD2])+
huffman@21164: apply (auto simp add: times_divide_eq_right [symmetric]
huffman@21164:             simp del: times_divide_eq)
huffman@21164: apply (drule_tac D = Db in lemma_nsderiv2, assumption+)
huffman@21164: apply (drule_tac
huffman@21164:      approx_minus_iff [THEN iffD2, THEN bex_Infinitesimal_iff2 [THEN iffD2]])
huffman@21164: apply (auto intro!: approx_add_mono1
huffman@21164:             simp add: left_distrib right_distrib mult_commute add_assoc)
huffman@21164: apply (rule_tac b1 = "hypreal_of_real Db * hypreal_of_real (f x)"
huffman@21164:          in add_commute [THEN subst])
huffman@21164: apply (auto intro!: Infinitesimal_add_approx_self2 [THEN approx_sym]
huffman@21164:                     Infinitesimal_add Infinitesimal_mult
huffman@21164:                     Infinitesimal_hypreal_of_real_mult
huffman@21164:                     Infinitesimal_hypreal_of_real_mult2
huffman@21164:           simp add: add_assoc [symmetric])
huffman@21164: done
huffman@21164: 
huffman@21164: text{*Multiplying by a constant*}
huffman@21164: lemma NSDERIV_cmult: "NSDERIV f x :> D
huffman@21164:       ==> NSDERIV (%x. c * f x) x :> c*D"
huffman@21164: apply (simp only: times_divide_eq_right [symmetric] NSDERIV_NSLIM_iff
huffman@21164:                   minus_mult_right right_diff_distrib [symmetric])
huffman@21164: apply (erule NSLIM_const [THEN NSLIM_mult])
huffman@21164: done
huffman@21164: 
huffman@21164: text{*Negation of function*}
huffman@21164: lemma NSDERIV_minus: "NSDERIV f x :> D ==> NSDERIV (%x. -(f x)) x :> -D"
huffman@21164: proof (simp add: NSDERIV_NSLIM_iff)
huffman@21164:   assume "(\<lambda>h. (f (x + h) - f x) / h) -- 0 --NS> D"
huffman@21164:   hence deriv: "(\<lambda>h. - ((f(x+h) - f x) / h)) -- 0 --NS> - D"
huffman@21164:     by (rule NSLIM_minus)
huffman@21164:   have "\<forall>h. - ((f (x + h) - f x) / h) = (- f (x + h) + f x) / h"
huffman@21164:     by (simp add: minus_divide_left)
huffman@21164:   with deriv
huffman@21164:   show "(\<lambda>h. (- f (x + h) + f x) / h) -- 0 --NS> - D" by simp
huffman@21164: qed
huffman@21164: 
huffman@21164: text{*Subtraction*}
huffman@21164: lemma NSDERIV_add_minus: "[| NSDERIV f x :> Da; NSDERIV g x :> Db |] ==> NSDERIV (%x. f x + -g x) x :> Da + -Db"
huffman@21164: by (blast dest: NSDERIV_add NSDERIV_minus)
huffman@21164: 
huffman@21164: lemma NSDERIV_diff:
huffman@21164:      "[| NSDERIV f x :> Da; NSDERIV g x :> Db |]
huffman@21164:       ==> NSDERIV (%x. f x - g x) x :> Da-Db"
huffman@21164: apply (simp add: diff_minus)
huffman@21164: apply (blast intro: NSDERIV_add_minus)
huffman@21164: done
huffman@21164: 
huffman@21164: text{*  Similarly to the above, the chain rule admits an entirely
huffman@21164:    straightforward derivation. Compare this with Harrison's
huffman@21164:    HOL proof of the chain rule, which proved to be trickier and
huffman@21164:    required an alternative characterisation of differentiability-
huffman@21164:    the so-called Carathedory derivative. Our main problem is
huffman@21164:    manipulation of terms.*}
huffman@21164: 
huffman@21164: 
huffman@21164: (* lemmas *)
huffman@21164: lemma NSDERIV_zero:
huffman@21164:       "[| NSDERIV g x :> D;
huffman@21164:                ( *f* g) (hypreal_of_real(x) + xa) = hypreal_of_real(g x);
huffman@21164:                xa \<in> Infinitesimal;
huffman@21164:                xa \<noteq> 0
huffman@21164:             |] ==> D = 0"
huffman@21164: apply (simp add: nsderiv_def)
huffman@21164: apply (drule bspec, auto)
huffman@21164: done
huffman@21164: 
huffman@21164: (* can be proved differently using NSLIM_isCont_iff *)
huffman@21164: lemma NSDERIV_approx:
huffman@21164:      "[| NSDERIV f x :> D;  h \<in> Infinitesimal;  h \<noteq> 0 |]
huffman@21164:       ==> ( *f* f) (hypreal_of_real(x) + h) - hypreal_of_real(f x) \<approx> 0"
huffman@21164: apply (simp add: nsderiv_def)
huffman@21164: apply (simp add: mem_infmal_iff [symmetric])
huffman@21164: apply (rule Infinitesimal_ratio)
huffman@21164: apply (rule_tac [3] approx_hypreal_of_real_HFinite, auto)
huffman@21164: done
huffman@21164: 
huffman@21164: (*---------------------------------------------------------------
huffman@21164:    from one version of differentiability
huffman@21164: 
huffman@21164:                 f(x) - f(a)
huffman@21164:               --------------- \<approx> Db
huffman@21164:                   x - a
huffman@21164:  ---------------------------------------------------------------*)
huffman@21164: lemma NSDERIVD1: "[| NSDERIV f (g x) :> Da;
huffman@21164:          ( *f* g) (hypreal_of_real(x) + xa) \<noteq> hypreal_of_real (g x);
huffman@21164:          ( *f* g) (hypreal_of_real(x) + xa) \<approx> hypreal_of_real (g x)
huffman@21164:       |] ==> (( *f* f) (( *f* g) (hypreal_of_real(x) + xa))
huffman@21164:                    - hypreal_of_real (f (g x)))
huffman@21164:               / (( *f* g) (hypreal_of_real(x) + xa) - hypreal_of_real (g x))
huffman@21164:              \<approx> hypreal_of_real(Da)"
huffman@21164: by (auto simp add: NSDERIV_NSLIM_iff2 NSLIM_def diff_minus [symmetric])
huffman@21164: 
huffman@21164: (*--------------------------------------------------------------
huffman@21164:    from other version of differentiability
huffman@21164: 
huffman@21164:                 f(x + h) - f(x)
huffman@21164:                ----------------- \<approx> Db
huffman@21164:                        h
huffman@21164:  --------------------------------------------------------------*)
huffman@21164: lemma NSDERIVD2: "[| NSDERIV g x :> Db; xa \<in> Infinitesimal; xa \<noteq> 0 |]
huffman@21164:       ==> (( *f* g) (hypreal_of_real(x) + xa) - hypreal_of_real(g x)) / xa
huffman@21164:           \<approx> hypreal_of_real(Db)"
huffman@21164: by (auto simp add: NSDERIV_NSLIM_iff NSLIM_def mem_infmal_iff starfun_lambda_cancel)
huffman@21164: 
huffman@21164: lemma lemma_chain: "(z::hypreal) \<noteq> 0 ==> x*y = (x*inverse(z))*(z*y)"
huffman@21164: by auto
huffman@21164: 
huffman@21164: text{*This proof uses both definitions of differentiability.*}
huffman@21164: lemma NSDERIV_chain: "[| NSDERIV f (g x) :> Da; NSDERIV g x :> Db |]
huffman@21164:       ==> NSDERIV (f o g) x :> Da * Db"
huffman@21164: apply (simp (no_asm_simp) add: NSDERIV_NSLIM_iff NSLIM_def
huffman@21164:                 mem_infmal_iff [symmetric])
huffman@21164: apply clarify
huffman@21164: apply (frule_tac f = g in NSDERIV_approx)
huffman@21164: apply (auto simp add: starfun_lambda_cancel2 starfun_o [symmetric])
huffman@21164: apply (case_tac "( *f* g) (hypreal_of_real (x) + xa) = hypreal_of_real (g x) ")
huffman@21164: apply (drule_tac g = g in NSDERIV_zero)
huffman@21164: apply (auto simp add: divide_inverse)
huffman@21164: apply (rule_tac z1 = "( *f* g) (hypreal_of_real (x) + xa) - hypreal_of_real (g x) " and y1 = "inverse xa" in lemma_chain [THEN ssubst])
huffman@21164: apply (erule hypreal_not_eq_minus_iff [THEN iffD1])
huffman@21164: apply (rule approx_mult_hypreal_of_real)
huffman@21164: apply (simp_all add: divide_inverse [symmetric])
huffman@21164: apply (blast intro: NSDERIVD1 approx_minus_iff [THEN iffD2])
huffman@21164: apply (blast intro: NSDERIVD2)
huffman@21164: done
huffman@21164: 
huffman@21164: text{*Differentiation of natural number powers*}
huffman@21164: lemma NSDERIV_Id [simp]: "NSDERIV (%x. x) x :> 1"
huffman@21164: by (simp add: NSDERIV_NSLIM_iff NSLIM_def divide_self del: divide_self_if)
huffman@21164: 
huffman@21164: lemma NSDERIV_cmult_Id [simp]: "NSDERIV (op * c) x :> c"
huffman@21164: by (cut_tac c = c and x = x in NSDERIV_Id [THEN NSDERIV_cmult], simp)
huffman@21164: 
huffman@21164: (*Can't get rid of x \<noteq> 0 because it isn't continuous at zero*)
huffman@21164: lemma NSDERIV_inverse:
huffman@21164:      "x \<noteq> 0 ==> NSDERIV (%x. inverse(x)) x :> (- (inverse x ^ Suc (Suc 0)))"
huffman@21164: apply (simp add: nsderiv_def)
huffman@21164: apply (rule ballI, simp, clarify)
huffman@21164: apply (frule (1) Infinitesimal_add_not_zero)
huffman@21164: apply (simp add: add_commute)
huffman@21164: (*apply (auto simp add: starfun_inverse_inverse realpow_two
huffman@21164:         simp del: minus_mult_left [symmetric] minus_mult_right [symmetric])*)
huffman@21164: apply (simp add: inverse_add inverse_mult_distrib [symmetric]
huffman@21164:               inverse_minus_eq [symmetric] add_ac mult_ac diff_def
huffman@21164:             del: inverse_mult_distrib inverse_minus_eq
huffman@21164:                  minus_mult_left [symmetric] minus_mult_right [symmetric])
huffman@21164: apply (simp (no_asm_simp) add: mult_assoc [symmetric] right_distrib
huffman@21164:             del: minus_mult_left [symmetric] minus_mult_right [symmetric])
huffman@21164: apply (rule_tac y = "inverse (- hypreal_of_real x * hypreal_of_real x)" in approx_trans)
huffman@21164: apply (rule inverse_add_Infinitesimal_approx2)
huffman@21164: apply (auto dest!: hypreal_of_real_HFinite_diff_Infinitesimal
huffman@21164:             simp add: inverse_minus_eq [symmetric] HFinite_minus_iff)
huffman@21164: apply (rule Infinitesimal_HFinite_mult2, auto)
huffman@21164: done
huffman@21164: 
huffman@21164: subsubsection {* Equivalence of NS and Standard definitions *}
huffman@21164: 
huffman@21164: lemma divideR_eq_divide [simp]: "x /# y = x / y"
huffman@21164: by (simp add: real_scaleR_def divide_inverse mult_commute)
huffman@21164: 
huffman@21164: text{*Now equivalence between NSDERIV and DERIV*}
huffman@21164: lemma NSDERIV_DERIV_iff: "(NSDERIV f x :> D) = (DERIV f x :> D)"
huffman@21164: by (simp add: deriv_def NSDERIV_NSLIM_iff LIM_NSLIM_iff)
huffman@21164: 
huffman@21164: (* let's do the standard proof though theorem *)
huffman@21164: (* LIM_mult2 follows from a NS proof          *)
huffman@21164: 
huffman@21164: lemma DERIV_cmult:
huffman@21164:   fixes f :: "real \<Rightarrow> 'a::real_normed_algebra" shows
huffman@21164:       "DERIV f x :> D ==> DERIV (%x. c * f x) x :> c*D"
huffman@21164: by (drule DERIV_mult' [OF DERIV_const], simp)
huffman@21164: 
huffman@21164: (* standard version *)
huffman@21164: lemma DERIV_chain: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (f o g) x :> Da * Db"
huffman@21164: by (drule (1) DERIV_chain', simp add: o_def real_scaleR_def mult_commute)
huffman@21164: 
huffman@21164: lemma DERIV_chain2: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (%x. f (g x)) x :> Da * Db"
huffman@21164: by (auto dest: DERIV_chain simp add: o_def)
huffman@21164: 
huffman@21164: (*derivative of linear multiplication*)
huffman@21164: lemma DERIV_cmult_Id [simp]: "DERIV (op * c) x :> c"
huffman@21164: by (cut_tac c = c and x = x in DERIV_Id [THEN DERIV_cmult], simp)
huffman@21164: 
huffman@21164: lemma DERIV_pow: "DERIV (%x. x ^ n) x :> real n * (x ^ (n - Suc 0))"
huffman@21164: apply (cut_tac DERIV_power [OF DERIV_Id])
huffman@21164: apply (simp add: real_scaleR_def real_of_nat_def)
huffman@21164: done
huffman@21164: 
huffman@21164: (* NS version *)
huffman@21164: lemma NSDERIV_pow: "NSDERIV (%x. x ^ n) x :> real n * (x ^ (n - Suc 0))"
huffman@21164: by (simp add: NSDERIV_DERIV_iff DERIV_pow)
huffman@21164: 
huffman@21164: text{*Power of -1*}
huffman@21164: 
huffman@21164: lemma DERIV_inverse: "x \<noteq> 0 ==> DERIV (%x. inverse(x)) x :> (-(inverse x ^ Suc (Suc 0)))"
huffman@21164: by (drule DERIV_inverse' [OF DERIV_Id], simp)
huffman@21164: 
huffman@21164: text{*Derivative of inverse*}
huffman@21164: lemma DERIV_inverse_fun: "[| DERIV f x :> d; f(x) \<noteq> 0 |]
huffman@21164:       ==> DERIV (%x. inverse(f x)::real) x :> (- (d * inverse(f(x) ^ Suc (Suc 0))))"
huffman@21164: by (drule (1) DERIV_inverse', simp add: mult_ac)
huffman@21164: 
huffman@21164: lemma NSDERIV_inverse_fun: "[| NSDERIV f x :> d; f(x) \<noteq> 0 |]
huffman@21164:       ==> NSDERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ Suc (Suc 0))))"
huffman@21164: by (simp add: NSDERIV_DERIV_iff DERIV_inverse_fun del: realpow_Suc)
huffman@21164: 
huffman@21164: text{*Derivative of quotient*}
huffman@21164: lemma DERIV_quotient: "[| DERIV f x :> d; DERIV g x :> e; g(x) \<noteq> 0 |]
huffman@21164:        ==> DERIV (%y. f(y) / (g y) :: real) x :> (d*g(x) - (e*f(x))) / (g(x) ^ Suc (Suc 0))"
huffman@21164: by (drule (2) DERIV_divide, simp add: mult_commute)
huffman@21164: 
huffman@21164: lemma NSDERIV_quotient: "[| NSDERIV f x :> d; DERIV g x :> e; g(x) \<noteq> 0 |]
huffman@21164:        ==> NSDERIV (%y. f(y) / (g y)) x :> (d*g(x)
huffman@21164:                             - (e*f(x))) / (g(x) ^ Suc (Suc 0))"
huffman@21164: by (simp add: NSDERIV_DERIV_iff DERIV_quotient del: realpow_Suc)
huffman@21164: 
huffman@21164: lemma CARAT_NSDERIV: "NSDERIV f x :> l ==>
huffman@21164:       \<exists>g. (\<forall>z. f z - f x = g z * (z-x)) & isNSCont g x & g x = l"
huffman@21164: by (auto simp add: NSDERIV_DERIV_iff isNSCont_isCont_iff CARAT_DERIV
huffman@21164:                    real_scaleR_def mult_commute)
huffman@21164: 
huffman@21164: lemma hypreal_eq_minus_iff3: "(x = y + z) = (x + -z = (y::hypreal))"
huffman@21164: by auto
huffman@21164: 
huffman@21164: lemma CARAT_DERIVD:
huffman@21164:   assumes all: "\<forall>z. f z - f x = g z * (z-x)"
huffman@21164:       and nsc: "isNSCont g x"
huffman@21164:   shows "NSDERIV f x :> g x"
huffman@21164: proof -
huffman@21164:   from nsc
huffman@21164:   have "\<forall>w. w \<noteq> hypreal_of_real x \<and> w \<approx> hypreal_of_real x \<longrightarrow>
huffman@21164:          ( *f* g) w * (w - hypreal_of_real x) / (w - hypreal_of_real x) \<approx>
huffman@21164:          hypreal_of_real (g x)"
huffman@21164:     by (simp add: diff_minus isNSCont_def)
huffman@21164:   thus ?thesis using all
huffman@21164:     by (simp add: NSDERIV_iff2 starfun_if_eq cong: if_cong)
huffman@21164: qed
huffman@21164: 
huffman@21164: subsubsection {* Differentiability predicate *}
huffman@21164: 
huffman@21164: lemma differentiableD: "f differentiable x ==> \<exists>D. DERIV f x :> D"
huffman@21164: by (simp add: differentiable_def)
huffman@21164: 
huffman@21164: lemma differentiableI: "DERIV f x :> D ==> f differentiable x"
huffman@21164: by (force simp add: differentiable_def)
huffman@21164: 
huffman@21164: lemma NSdifferentiableD: "f NSdifferentiable x ==> \<exists>D. NSDERIV f x :> D"
huffman@21164: by (simp add: NSdifferentiable_def)
huffman@21164: 
huffman@21164: lemma NSdifferentiableI: "NSDERIV f x :> D ==> f NSdifferentiable x"
huffman@21164: by (force simp add: NSdifferentiable_def)
huffman@21164: 
huffman@21164: lemma differentiable_const: "(\<lambda>z. a) differentiable x"
huffman@21164:   apply (unfold differentiable_def)
huffman@21164:   apply (rule_tac x=0 in exI)
huffman@21164:   apply simp
huffman@21164:   done
huffman@21164: 
huffman@21164: lemma differentiable_sum:
huffman@21164:   assumes "f differentiable x"
huffman@21164:   and "g differentiable x"
huffman@21164:   shows "(\<lambda>x. f x + g x) differentiable x"
huffman@21164: proof -
huffman@21164:   from prems have "\<exists>D. DERIV f x :> D" by (unfold differentiable_def)
huffman@21164:   then obtain df where "DERIV f x :> df" ..
huffman@21164:   moreover from prems have "\<exists>D. DERIV g x :> D" by (unfold differentiable_def)
huffman@21164:   then obtain dg where "DERIV g x :> dg" ..
huffman@21164:   ultimately have "DERIV (\<lambda>x. f x + g x) x :> df + dg" by (rule DERIV_add)
huffman@21164:   hence "\<exists>D. DERIV (\<lambda>x. f x + g x) x :> D" by auto
huffman@21164:   thus ?thesis by (fold differentiable_def)
huffman@21164: qed
huffman@21164: 
huffman@21164: lemma differentiable_diff:
huffman@21164:   assumes "f differentiable x"
huffman@21164:   and "g differentiable x"
huffman@21164:   shows "(\<lambda>x. f x - g x) differentiable x"
huffman@21164: proof -
huffman@21164:   from prems have "f differentiable x" by simp
huffman@21164:   moreover
huffman@21164:   from prems have "\<exists>D. DERIV g x :> D" by (unfold differentiable_def)
huffman@21164:   then obtain dg where "DERIV g x :> dg" ..
huffman@21164:   then have "DERIV (\<lambda>x. - g x) x :> -dg" by (rule DERIV_minus)
huffman@21164:   hence "\<exists>D. DERIV (\<lambda>x. - g x) x :> D" by auto
huffman@21164:   hence "(\<lambda>x. - g x) differentiable x" by (fold differentiable_def)
huffman@21164:   ultimately 
huffman@21164:   show ?thesis
huffman@21164:     by (auto simp: real_diff_def dest: differentiable_sum)
huffman@21164: qed
huffman@21164: 
huffman@21164: lemma differentiable_mult:
huffman@21164:   assumes "f differentiable x"
huffman@21164:   and "g differentiable x"
huffman@21164:   shows "(\<lambda>x. f x * g x) differentiable x"
huffman@21164: proof -
huffman@21164:   from prems have "\<exists>D. DERIV f x :> D" by (unfold differentiable_def)
huffman@21164:   then obtain df where "DERIV f x :> df" ..
huffman@21164:   moreover from prems have "\<exists>D. DERIV g x :> D" by (unfold differentiable_def)
huffman@21164:   then obtain dg where "DERIV g x :> dg" ..
huffman@21164:   ultimately have "DERIV (\<lambda>x. f x * g x) x :> df * g x + dg * f x" by (simp add: DERIV_mult)
huffman@21164:   hence "\<exists>D. DERIV (\<lambda>x. f x * g x) x :> D" by auto
huffman@21164:   thus ?thesis by (fold differentiable_def)
huffman@21164: qed
huffman@21164: 
huffman@21164: subsection {*(NS) Increment*}
huffman@21164: lemma incrementI:
huffman@21164:       "f NSdifferentiable x ==>
huffman@21164:       increment f x h = ( *f* f) (hypreal_of_real(x) + h) -
huffman@21164:       hypreal_of_real (f x)"
huffman@21164: by (simp add: increment_def)
huffman@21164: 
huffman@21164: lemma incrementI2: "NSDERIV f x :> D ==>
huffman@21164:      increment f x h = ( *f* f) (hypreal_of_real(x) + h) -
huffman@21164:      hypreal_of_real (f x)"
huffman@21164: apply (erule NSdifferentiableI [THEN incrementI])
huffman@21164: done
huffman@21164: 
huffman@21164: (* The Increment theorem -- Keisler p. 65 *)
huffman@21164: lemma increment_thm: "[| NSDERIV f x :> D; h \<in> Infinitesimal; h \<noteq> 0 |]
huffman@21164:       ==> \<exists>e \<in> Infinitesimal. increment f x h = hypreal_of_real(D)*h + e*h"
huffman@21164: apply (frule_tac h = h in incrementI2, simp add: nsderiv_def)
huffman@21164: apply (drule bspec, auto)
huffman@21164: apply (drule bex_Infinitesimal_iff2 [THEN iffD2], clarify)
huffman@21164: apply (frule_tac b1 = "hypreal_of_real (D) + y"
huffman@21164:         in hypreal_mult_right_cancel [THEN iffD2])
huffman@21164: apply (erule_tac [2] V = "(( *f* f) (hypreal_of_real (x) + h) - hypreal_of_real (f x)) / h = hypreal_of_real (D) + y" in thin_rl)
huffman@21164: apply assumption
huffman@21164: apply (simp add: times_divide_eq_right [symmetric])
huffman@21164: apply (auto simp add: left_distrib)
huffman@21164: done
huffman@21164: 
huffman@21164: lemma increment_thm2:
huffman@21164:      "[| NSDERIV f x :> D; h \<approx> 0; h \<noteq> 0 |]
huffman@21164:       ==> \<exists>e \<in> Infinitesimal. increment f x h =
huffman@21164:               hypreal_of_real(D)*h + e*h"
huffman@21164: by (blast dest!: mem_infmal_iff [THEN iffD2] intro!: increment_thm)
huffman@21164: 
huffman@21164: 
huffman@21164: lemma increment_approx_zero: "[| NSDERIV f x :> D; h \<approx> 0; h \<noteq> 0 |]
huffman@21164:       ==> increment f x h \<approx> 0"
huffman@21164: apply (drule increment_thm2,
huffman@21164:        auto intro!: Infinitesimal_HFinite_mult2 HFinite_add simp add: left_distrib [symmetric] mem_infmal_iff [symmetric])
huffman@21164: apply (erule Infinitesimal_subset_HFinite [THEN subsetD])
huffman@21164: done
huffman@21164: 
huffman@21164: subsection {* Nested Intervals and Bisection *}
huffman@21164: 
huffman@21164: text{*Lemmas about nested intervals and proof by bisection (cf.Harrison).
huffman@21164:      All considerably tidied by lcp.*}
huffman@21164: 
huffman@21164: 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)"
huffman@21164: apply (induct "no")
huffman@21164: apply (auto intro: order_trans)
huffman@21164: done
huffman@21164: 
huffman@21164: lemma f_inc_g_dec_Beq_f: "[| \<forall>n. f(n) \<le> f(Suc n);
huffman@21164:          \<forall>n. g(Suc n) \<le> g(n);
huffman@21164:          \<forall>n. f(n) \<le> g(n) |]
huffman@21164:       ==> Bseq (f :: nat \<Rightarrow> real)"
huffman@21164: apply (rule_tac k = "f 0" and K = "g 0" in BseqI2, rule allI)
huffman@21164: apply (induct_tac "n")
huffman@21164: apply (auto intro: order_trans)
huffman@21164: apply (rule_tac y = "g (Suc na)" in order_trans)
huffman@21164: apply (induct_tac [2] "na")
huffman@21164: apply (auto intro: order_trans)
huffman@21164: done
huffman@21164: 
huffman@21164: lemma f_inc_g_dec_Beq_g: "[| \<forall>n. f(n) \<le> f(Suc n);
huffman@21164:          \<forall>n. g(Suc n) \<le> g(n);
huffman@21164:          \<forall>n. f(n) \<le> g(n) |]
huffman@21164:       ==> Bseq (g :: nat \<Rightarrow> real)"
huffman@21164: apply (subst Bseq_minus_iff [symmetric])
huffman@21164: apply (rule_tac g = "%x. - (f x)" in f_inc_g_dec_Beq_f)
huffman@21164: apply auto
huffman@21164: done
huffman@21164: 
huffman@21164: lemma f_inc_imp_le_lim:
huffman@21164:   fixes f :: "nat \<Rightarrow> real"
huffman@21164:   shows "\<lbrakk>\<forall>n. f n \<le> f (Suc n); convergent f\<rbrakk> \<Longrightarrow> f n \<le> lim f"
huffman@21164: apply (rule linorder_not_less [THEN iffD1])
huffman@21164: apply (auto simp add: convergent_LIMSEQ_iff LIMSEQ_iff monoseq_Suc)
huffman@21164: apply (drule real_less_sum_gt_zero)
huffman@21164: apply (drule_tac x = "f n + - lim f" in spec, safe)
huffman@21164: apply (drule_tac P = "%na. no\<le>na --> ?Q na" and x = "no + n" in spec, auto)
huffman@21164: apply (subgoal_tac "lim f \<le> f (no + n) ")
huffman@21164: apply (drule_tac no=no and m=n in lemma_f_mono_add)
huffman@21164: apply (auto simp add: add_commute)
huffman@21164: apply (induct_tac "no")
huffman@21164: apply simp
huffman@21164: apply (auto intro: order_trans simp add: diff_minus abs_if)
huffman@21164: done
huffman@21164: 
huffman@21164: lemma lim_uminus: "convergent g ==> lim (%x. - g x) = - (lim g)"
huffman@21164: apply (rule LIMSEQ_minus [THEN limI])
huffman@21164: apply (simp add: convergent_LIMSEQ_iff)
huffman@21164: done
huffman@21164: 
huffman@21164: lemma g_dec_imp_lim_le:
huffman@21164:   fixes g :: "nat \<Rightarrow> real"
huffman@21164:   shows "\<lbrakk>\<forall>n. g (Suc n) \<le> g(n); convergent g\<rbrakk> \<Longrightarrow> lim g \<le> g n"
huffman@21164: apply (subgoal_tac "- (g n) \<le> - (lim g) ")
huffman@21164: apply (cut_tac [2] f = "%x. - (g x)" in f_inc_imp_le_lim)
huffman@21164: apply (auto simp add: lim_uminus convergent_minus_iff [symmetric])
huffman@21164: done
huffman@21164: 
huffman@21164: lemma lemma_nest: "[| \<forall>n. f(n) \<le> f(Suc n);
huffman@21164:          \<forall>n. g(Suc n) \<le> g(n);
huffman@21164:          \<forall>n. f(n) \<le> g(n) |]
huffman@21164:       ==> \<exists>l m :: real. l \<le> m &  ((\<forall>n. f(n) \<le> l) & f ----> l) &
huffman@21164:                             ((\<forall>n. m \<le> g(n)) & g ----> m)"
huffman@21164: apply (subgoal_tac "monoseq f & monoseq g")
huffman@21164: prefer 2 apply (force simp add: LIMSEQ_iff monoseq_Suc)
huffman@21164: apply (subgoal_tac "Bseq f & Bseq g")
huffman@21164: prefer 2 apply (blast intro: f_inc_g_dec_Beq_f f_inc_g_dec_Beq_g)
huffman@21164: apply (auto dest!: Bseq_monoseq_convergent simp add: convergent_LIMSEQ_iff)
huffman@21164: apply (rule_tac x = "lim f" in exI)
huffman@21164: apply (rule_tac x = "lim g" in exI)
huffman@21164: apply (auto intro: LIMSEQ_le)
huffman@21164: apply (auto simp add: f_inc_imp_le_lim g_dec_imp_lim_le convergent_LIMSEQ_iff)
huffman@21164: done
huffman@21164: 
huffman@21164: lemma lemma_nest_unique: "[| \<forall>n. f(n) \<le> f(Suc n);
huffman@21164:          \<forall>n. g(Suc n) \<le> g(n);
huffman@21164:          \<forall>n. f(n) \<le> g(n);
huffman@21164:          (%n. f(n) - g(n)) ----> 0 |]
huffman@21164:       ==> \<exists>l::real. ((\<forall>n. f(n) \<le> l) & f ----> l) &
huffman@21164:                 ((\<forall>n. l \<le> g(n)) & g ----> l)"
huffman@21164: apply (drule lemma_nest, auto)
huffman@21164: apply (subgoal_tac "l = m")
huffman@21164: apply (drule_tac [2] X = f in LIMSEQ_diff)
huffman@21164: apply (auto intro: LIMSEQ_unique)
huffman@21164: done
huffman@21164: 
huffman@21164: text{*The universal quantifiers below are required for the declaration
huffman@21164:   of @{text Bolzano_nest_unique} below.*}
huffman@21164: 
huffman@21164: lemma Bolzano_bisect_le:
huffman@21164:  "a \<le> b ==> \<forall>n. fst (Bolzano_bisect P a b n) \<le> snd (Bolzano_bisect P a b n)"
huffman@21164: apply (rule allI)
huffman@21164: apply (induct_tac "n")
huffman@21164: apply (auto simp add: Let_def split_def)
huffman@21164: done
huffman@21164: 
huffman@21164: lemma Bolzano_bisect_fst_le_Suc: "a \<le> b ==>
huffman@21164:    \<forall>n. fst(Bolzano_bisect P a b n) \<le> fst (Bolzano_bisect P a b (Suc n))"
huffman@21164: apply (rule allI)
huffman@21164: apply (induct_tac "n")
huffman@21164: apply (auto simp add: Bolzano_bisect_le Let_def split_def)
huffman@21164: done
huffman@21164: 
huffman@21164: lemma Bolzano_bisect_Suc_le_snd: "a \<le> b ==>
huffman@21164:    \<forall>n. snd(Bolzano_bisect P a b (Suc n)) \<le> snd (Bolzano_bisect P a b n)"
huffman@21164: apply (rule allI)
huffman@21164: apply (induct_tac "n")
huffman@21164: apply (auto simp add: Bolzano_bisect_le Let_def split_def)
huffman@21164: done
huffman@21164: 
huffman@21164: lemma eq_divide_2_times_iff: "((x::real) = y / (2 * z)) = (2 * x = y/z)"
huffman@21164: apply (auto)
huffman@21164: apply (drule_tac f = "%u. (1/2) *u" in arg_cong)
huffman@21164: apply (simp)
huffman@21164: done
huffman@21164: 
huffman@21164: lemma Bolzano_bisect_diff:
huffman@21164:      "a \<le> b ==>
huffman@21164:       snd(Bolzano_bisect P a b n) - fst(Bolzano_bisect P a b n) =
huffman@21164:       (b-a) / (2 ^ n)"
huffman@21164: apply (induct "n")
huffman@21164: apply (auto simp add: eq_divide_2_times_iff add_divide_distrib Let_def split_def)
huffman@21164: done
huffman@21164: 
huffman@21164: lemmas Bolzano_nest_unique =
huffman@21164:     lemma_nest_unique
huffman@21164:     [OF Bolzano_bisect_fst_le_Suc Bolzano_bisect_Suc_le_snd Bolzano_bisect_le]
huffman@21164: 
huffman@21164: 
huffman@21164: lemma not_P_Bolzano_bisect:
huffman@21164:   assumes P:    "!!a b c. [| P(a,b); P(b,c); a \<le> b; b \<le> c|] ==> P(a,c)"
huffman@21164:       and notP: "~ P(a,b)"
huffman@21164:       and le:   "a \<le> b"
huffman@21164:   shows "~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))"
huffman@21164: proof (induct n)
huffman@21164:   case 0 thus ?case by simp
huffman@21164:  next
huffman@21164:   case (Suc n)
huffman@21164:   thus ?case
huffman@21164:  by (auto simp del: surjective_pairing [symmetric]
huffman@21164:              simp add: Let_def split_def Bolzano_bisect_le [OF le]
huffman@21164:      P [of "fst (Bolzano_bisect P a b n)" _ "snd (Bolzano_bisect P a b n)"])
huffman@21164: qed
huffman@21164: 
huffman@21164: (*Now we re-package P_prem as a formula*)
huffman@21164: lemma not_P_Bolzano_bisect':
huffman@21164:      "[| \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c --> P(a,c);
huffman@21164:          ~ P(a,b);  a \<le> b |] ==>
huffman@21164:       \<forall>n. ~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))"
huffman@21164: by (blast elim!: not_P_Bolzano_bisect [THEN [2] rev_notE])
huffman@21164: 
huffman@21164: 
huffman@21164: 
huffman@21164: lemma lemma_BOLZANO:
huffman@21164:      "[| \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c --> P(a,c);
huffman@21164:          \<forall>x. \<exists>d::real. 0 < d &
huffman@21164:                 (\<forall>a b. a \<le> x & x \<le> b & (b-a) < d --> P(a,b));
huffman@21164:          a \<le> b |]
huffman@21164:       ==> P(a,b)"
huffman@21164: apply (rule Bolzano_nest_unique [where P1=P, THEN exE], assumption+)
huffman@21164: apply (rule LIMSEQ_minus_cancel)
huffman@21164: apply (simp (no_asm_simp) add: Bolzano_bisect_diff LIMSEQ_divide_realpow_zero)
huffman@21164: apply (rule ccontr)
huffman@21164: apply (drule not_P_Bolzano_bisect', assumption+)
huffman@21164: apply (rename_tac "l")
huffman@21164: apply (drule_tac x = l in spec, clarify)
huffman@21164: apply (simp add: LIMSEQ_def)
huffman@21164: apply (drule_tac P = "%r. 0<r --> ?Q r" and x = "d/2" in spec)
huffman@21164: apply (drule_tac P = "%r. 0<r --> ?Q r" and x = "d/2" in spec)
huffman@21164: apply (drule real_less_half_sum, auto)
huffman@21164: apply (drule_tac x = "fst (Bolzano_bisect P a b (no + noa))" in spec)
huffman@21164: apply (drule_tac x = "snd (Bolzano_bisect P a b (no + noa))" in spec)
huffman@21164: apply safe
huffman@21164: apply (simp_all (no_asm_simp))
huffman@21164: 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)
huffman@21164: apply (simp (no_asm_simp) add: abs_if)
huffman@21164: apply (rule real_sum_of_halves [THEN subst])
huffman@21164: apply (rule add_strict_mono)
huffman@21164: apply (simp_all add: diff_minus [symmetric])
huffman@21164: done
huffman@21164: 
huffman@21164: 
huffman@21164: lemma lemma_BOLZANO2: "((\<forall>a b c. (a \<le> b & b \<le> c & P(a,b) & P(b,c)) --> P(a,c)) &
huffman@21164:        (\<forall>x. \<exists>d::real. 0 < d &
huffman@21164:                 (\<forall>a b. a \<le> x & x \<le> b & (b-a) < d --> P(a,b))))
huffman@21164:       --> (\<forall>a b. a \<le> b --> P(a,b))"
huffman@21164: apply clarify
huffman@21164: apply (blast intro: lemma_BOLZANO)
huffman@21164: done
huffman@21164: 
huffman@21164: 
huffman@21164: subsection {* Intermediate Value Theorem *}
huffman@21164: 
huffman@21164: text {*Prove Contrapositive by Bisection*}
huffman@21164: 
huffman@21164: lemma IVT: "[| f(a::real) \<le> (y::real); y \<le> f(b);
huffman@21164:          a \<le> b;
huffman@21164:          (\<forall>x. a \<le> x & x \<le> b --> isCont f x) |]
huffman@21164:       ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y"
huffman@21164: apply (rule contrapos_pp, assumption)
huffman@21164: 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)
huffman@21164: apply safe
huffman@21164: apply simp_all
huffman@21164: apply (simp add: isCont_iff LIM_def)
huffman@21164: apply (rule ccontr)
huffman@21164: apply (subgoal_tac "a \<le> x & x \<le> b")
huffman@21164:  prefer 2
huffman@21164:  apply simp
huffman@21164:  apply (drule_tac P = "%d. 0<d --> ?P d" and x = 1 in spec, arith)
huffman@21164: apply (drule_tac x = x in spec)+
huffman@21164: apply simp
huffman@21164: apply (drule_tac P = "%r. ?P r --> (\<exists>s>0. ?Q r s) " and x = "\<bar>y - f x\<bar>" in spec)
huffman@21164: apply safe
huffman@21164: apply simp
huffman@21164: apply (drule_tac x = s in spec, clarify)
huffman@21164: apply (cut_tac x = "f x" and y = y in linorder_less_linear, safe)
huffman@21164: apply (drule_tac x = "ba-x" in spec)
huffman@21164: apply (simp_all add: abs_if)
huffman@21164: apply (drule_tac x = "aa-x" in spec)
huffman@21164: apply (case_tac "x \<le> aa", simp_all)
huffman@21164: done
huffman@21164: 
huffman@21164: lemma IVT2: "[| f(b::real) \<le> (y::real); y \<le> f(a);
huffman@21164:          a \<le> b;
huffman@21164:          (\<forall>x. a \<le> x & x \<le> b --> isCont f x)
huffman@21164:       |] ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y"
huffman@21164: apply (subgoal_tac "- f a \<le> -y & -y \<le> - f b", clarify)
huffman@21164: apply (drule IVT [where f = "%x. - f x"], assumption)
huffman@21164: apply (auto intro: isCont_minus)
huffman@21164: done
huffman@21164: 
huffman@21164: (*HOL style here: object-level formulations*)
huffman@21164: lemma IVT_objl: "(f(a::real) \<le> (y::real) & y \<le> f(b) & a \<le> b &
huffman@21164:       (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
huffman@21164:       --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
huffman@21164: apply (blast intro: IVT)
huffman@21164: done
huffman@21164: 
huffman@21164: lemma IVT2_objl: "(f(b::real) \<le> (y::real) & y \<le> f(a) & a \<le> b &
huffman@21164:       (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
huffman@21164:       --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
huffman@21164: apply (blast intro: IVT2)
huffman@21164: done
huffman@21164: 
huffman@21164: text{*By bisection, function continuous on closed interval is bounded above*}
huffman@21164: 
huffman@21164: lemma isCont_bounded:
huffman@21164:      "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
huffman@21164:       ==> \<exists>M::real. \<forall>x::real. a \<le> x & x \<le> b --> f(x) \<le> M"
huffman@21164: 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)
huffman@21164: apply safe
huffman@21164: apply simp_all
huffman@21164: apply (rename_tac x xa ya M Ma)
huffman@21164: apply (cut_tac x = M and y = Ma in linorder_linear, safe)
huffman@21164: apply (rule_tac x = Ma in exI, clarify)
huffman@21164: apply (cut_tac x = xb and y = xa in linorder_linear, force)
huffman@21164: apply (rule_tac x = M in exI, clarify)
huffman@21164: apply (cut_tac x = xb and y = xa in linorder_linear, force)
huffman@21164: apply (case_tac "a \<le> x & x \<le> b")
huffman@21164: apply (rule_tac [2] x = 1 in exI)
huffman@21164: prefer 2 apply force
huffman@21164: apply (simp add: LIM_def isCont_iff)
huffman@21164: apply (drule_tac x = x in spec, auto)
huffman@21164: apply (erule_tac V = "\<forall>M. \<exists>x. a \<le> x & x \<le> b & ~ f x \<le> M" in thin_rl)
huffman@21164: apply (drule_tac x = 1 in spec, auto)
huffman@21164: apply (rule_tac x = s in exI, clarify)
huffman@21164: apply (rule_tac x = "\<bar>f x\<bar> + 1" in exI, clarify)
huffman@21164: apply (drule_tac x = "xa-x" in spec)
huffman@21164: apply (auto simp add: abs_ge_self)
huffman@21164: done
huffman@21164: 
huffman@21164: text{*Refine the above to existence of least upper bound*}
huffman@21164: 
huffman@21164: lemma lemma_reals_complete: "((\<exists>x. x \<in> S) & (\<exists>y. isUb UNIV S (y::real))) -->
huffman@21164:       (\<exists>t. isLub UNIV S t)"
huffman@21164: by (blast intro: reals_complete)
huffman@21164: 
huffman@21164: lemma isCont_has_Ub: "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
huffman@21164:          ==> \<exists>M::real. (\<forall>x::real. a \<le> x & x \<le> b --> f(x) \<le> M) &
huffman@21164:                    (\<forall>N. N < M --> (\<exists>x. a \<le> x & x \<le> b & N < f(x)))"
huffman@21164: apply (cut_tac S = "Collect (%y. \<exists>x. a \<le> x & x \<le> b & y = f x)"
huffman@21164:         in lemma_reals_complete)
huffman@21164: apply auto
huffman@21164: apply (drule isCont_bounded, assumption)
huffman@21164: apply (auto simp add: isUb_def leastP_def isLub_def setge_def setle_def)
huffman@21164: apply (rule exI, auto)
huffman@21164: apply (auto dest!: spec simp add: linorder_not_less)
huffman@21164: done
huffman@21164: 
huffman@21164: text{*Now show that it attains its upper bound*}
huffman@21164: 
huffman@21164: lemma isCont_eq_Ub:
huffman@21164:   assumes le: "a \<le> b"
huffman@21164:       and con: "\<forall>x::real. a \<le> x & x \<le> b --> isCont f x"
huffman@21164:   shows "\<exists>M::real. (\<forall>x. a \<le> x & x \<le> b --> f(x) \<le> M) &
huffman@21164:              (\<exists>x. a \<le> x & x \<le> b & f(x) = M)"
huffman@21164: proof -
huffman@21164:   from isCont_has_Ub [OF le con]
huffman@21164:   obtain M where M1: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M"
huffman@21164:              and M2: "!!N. N<M ==> \<exists>x. a \<le> x \<and> x \<le> b \<and> N < f x"  by blast
huffman@21164:   show ?thesis
huffman@21164:   proof (intro exI, intro conjI)
huffman@21164:     show " \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M" by (rule M1)
huffman@21164:     show "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M"
huffman@21164:     proof (rule ccontr)
huffman@21164:       assume "\<not> (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M)"
huffman@21164:       with M1 have M3: "\<forall>x. a \<le> x & x \<le> b --> f x < M"
huffman@21164:         by (fastsimp simp add: linorder_not_le [symmetric])
huffman@21164:       hence "\<forall>x. a \<le> x & x \<le> b --> isCont (%x. inverse (M - f x)) x"
huffman@21164:         by (auto simp add: isCont_inverse isCont_diff con)
huffman@21164:       from isCont_bounded [OF le this]
huffman@21164:       obtain k where k: "!!x. a \<le> x & x \<le> b --> inverse (M - f x) \<le> k" by auto
huffman@21164:       have Minv: "!!x. a \<le> x & x \<le> b --> 0 < inverse (M - f (x))"
huffman@21164:         by (simp add: M3 compare_rls)
huffman@21164:       have "!!x. a \<le> x & x \<le> b --> inverse (M - f x) < k+1" using k
huffman@21164:         by (auto intro: order_le_less_trans [of _ k])
huffman@21164:       with Minv
huffman@21164:       have "!!x. a \<le> x & x \<le> b --> inverse(k+1) < inverse(inverse(M - f x))"
huffman@21164:         by (intro strip less_imp_inverse_less, simp_all)
huffman@21164:       hence invlt: "!!x. a \<le> x & x \<le> b --> inverse(k+1) < M - f x"
huffman@21164:         by simp
huffman@21164:       have "M - inverse (k+1) < M" using k [of a] Minv [of a] le
huffman@21164:         by (simp, arith)
huffman@21164:       from M2 [OF this]
huffman@21164:       obtain x where ax: "a \<le> x & x \<le> b & M - inverse(k+1) < f x" ..
huffman@21164:       thus False using invlt [of x] by force
huffman@21164:     qed
huffman@21164:   qed
huffman@21164: qed
huffman@21164: 
huffman@21164: 
huffman@21164: text{*Same theorem for lower bound*}
huffman@21164: 
huffman@21164: lemma isCont_eq_Lb: "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
huffman@21164:          ==> \<exists>M::real. (\<forall>x::real. a \<le> x & x \<le> b --> M \<le> f(x)) &
huffman@21164:                    (\<exists>x. a \<le> x & x \<le> b & f(x) = M)"
huffman@21164: apply (subgoal_tac "\<forall>x. a \<le> x & x \<le> b --> isCont (%x. - (f x)) x")
huffman@21164: prefer 2 apply (blast intro: isCont_minus)
huffman@21164: apply (drule_tac f = "(%x. - (f x))" in isCont_eq_Ub)
huffman@21164: apply safe
huffman@21164: apply auto
huffman@21164: done
huffman@21164: 
huffman@21164: 
huffman@21164: text{*Another version.*}
huffman@21164: 
huffman@21164: lemma isCont_Lb_Ub: "[|a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
huffman@21164:       ==> \<exists>L M::real. (\<forall>x::real. a \<le> x & x \<le> b --> L \<le> f(x) & f(x) \<le> M) &
huffman@21164:           (\<forall>y. L \<le> y & y \<le> M --> (\<exists>x. a \<le> x & x \<le> b & (f(x) = y)))"
huffman@21164: apply (frule isCont_eq_Lb)
huffman@21164: apply (frule_tac [2] isCont_eq_Ub)
huffman@21164: apply (assumption+, safe)
huffman@21164: apply (rule_tac x = "f x" in exI)
huffman@21164: apply (rule_tac x = "f xa" in exI, simp, safe)
huffman@21164: apply (cut_tac x = x and y = xa in linorder_linear, safe)
huffman@21164: apply (cut_tac f = f and a = x and b = xa and y = y in IVT_objl)
huffman@21164: apply (cut_tac [2] f = f and a = xa and b = x and y = y in IVT2_objl, safe)
huffman@21164: apply (rule_tac [2] x = xb in exI)
huffman@21164: apply (rule_tac [4] x = xb in exI, simp_all)
huffman@21164: done
huffman@21164: 
huffman@21164: 
huffman@21164: text{*If @{term "0 < f'(x)"} then @{term x} is Locally Strictly Increasing At The Right*}
huffman@21164: 
huffman@21164: lemma DERIV_left_inc:
huffman@21164:   fixes f :: "real => real"
huffman@21164:   assumes der: "DERIV f x :> l"
huffman@21164:       and l:   "0 < l"
huffman@21164:   shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x + h)"
huffman@21164: proof -
huffman@21164:   from l der [THEN DERIV_D, THEN LIM_D [where r = "l"]]
huffman@21164:   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)"
huffman@21164:     by (simp add: diff_minus)
huffman@21164:   then obtain s
huffman@21164:         where s:   "0 < s"
huffman@21164:           and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < l"
huffman@21164:     by auto
huffman@21164:   thus ?thesis
huffman@21164:   proof (intro exI conjI strip)
huffman@21164:     show "0<s" .
huffman@21164:     fix h::real
huffman@21164:     assume "0 < h" "h < s"
huffman@21164:     with all [of h] show "f x < f (x+h)"
huffman@21164:     proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric]
huffman@21164:     split add: split_if_asm)
huffman@21164:       assume "~ (f (x+h) - f x) / h < l" and h: "0 < h"
huffman@21164:       with l
huffman@21164:       have "0 < (f (x+h) - f x) / h" by arith
huffman@21164:       thus "f x < f (x+h)"
huffman@21164:   by (simp add: pos_less_divide_eq h)
huffman@21164:     qed
huffman@21164:   qed
huffman@21164: qed
huffman@21164: 
huffman@21164: lemma DERIV_left_dec:
huffman@21164:   fixes f :: "real => real"
huffman@21164:   assumes der: "DERIV f x :> l"
huffman@21164:       and l:   "l < 0"
huffman@21164:   shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x-h)"
huffman@21164: proof -
huffman@21164:   from l der [THEN DERIV_D, THEN LIM_D [where r = "-l"]]
huffman@21164:   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)"
huffman@21164:     by (simp add: diff_minus)
huffman@21164:   then obtain s
huffman@21164:         where s:   "0 < s"
huffman@21164:           and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < -l"
huffman@21164:     by auto
huffman@21164:   thus ?thesis
huffman@21164:   proof (intro exI conjI strip)
huffman@21164:     show "0<s" .
huffman@21164:     fix h::real
huffman@21164:     assume "0 < h" "h < s"
huffman@21164:     with all [of "-h"] show "f x < f (x-h)"
huffman@21164:     proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric]
huffman@21164:     split add: split_if_asm)
huffman@21164:       assume " - ((f (x-h) - f x) / h) < l" and h: "0 < h"
huffman@21164:       with l
huffman@21164:       have "0 < (f (x-h) - f x) / h" by arith
huffman@21164:       thus "f x < f (x-h)"
huffman@21164:   by (simp add: pos_less_divide_eq h)
huffman@21164:     qed
huffman@21164:   qed
huffman@21164: qed
huffman@21164: 
huffman@21164: lemma DERIV_local_max:
huffman@21164:   fixes f :: "real => real"
huffman@21164:   assumes der: "DERIV f x :> l"
huffman@21164:       and d:   "0 < d"
huffman@21164:       and le:  "\<forall>y. \<bar>x-y\<bar> < d --> f(y) \<le> f(x)"
huffman@21164:   shows "l = 0"
huffman@21164: proof (cases rule: linorder_cases [of l 0])
huffman@21164:   case equal show ?thesis .
huffman@21164: next
huffman@21164:   case less
huffman@21164:   from DERIV_left_dec [OF der less]
huffman@21164:   obtain d' where d': "0 < d'"
huffman@21164:              and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x-h)" by blast
huffman@21164:   from real_lbound_gt_zero [OF d d']
huffman@21164:   obtain e where "0 < e \<and> e < d \<and> e < d'" ..
huffman@21164:   with lt le [THEN spec [where x="x-e"]]
huffman@21164:   show ?thesis by (auto simp add: abs_if)
huffman@21164: next
huffman@21164:   case greater
huffman@21164:   from DERIV_left_inc [OF der greater]
huffman@21164:   obtain d' where d': "0 < d'"
huffman@21164:              and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x + h)" by blast
huffman@21164:   from real_lbound_gt_zero [OF d d']
huffman@21164:   obtain e where "0 < e \<and> e < d \<and> e < d'" ..
huffman@21164:   with lt le [THEN spec [where x="x+e"]]
huffman@21164:   show ?thesis by (auto simp add: abs_if)
huffman@21164: qed
huffman@21164: 
huffman@21164: 
huffman@21164: text{*Similar theorem for a local minimum*}
huffman@21164: lemma DERIV_local_min:
huffman@21164:   fixes f :: "real => real"
huffman@21164:   shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) \<le> f(y) |] ==> l = 0"
huffman@21164: by (drule DERIV_minus [THEN DERIV_local_max], auto)
huffman@21164: 
huffman@21164: 
huffman@21164: text{*In particular, if a function is locally flat*}
huffman@21164: lemma DERIV_local_const:
huffman@21164:   fixes f :: "real => real"
huffman@21164:   shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) = f(y) |] ==> l = 0"
huffman@21164: by (auto dest!: DERIV_local_max)
huffman@21164: 
huffman@21164: text{*Lemma about introducing open ball in open interval*}
huffman@21164: lemma lemma_interval_lt:
huffman@21164:      "[| a < x;  x < b |]
huffman@21164:       ==> \<exists>d::real. 0 < d & (\<forall>y. \<bar>x-y\<bar> < d --> a < y & y < b)"
huffman@21164: apply (simp add: abs_interval_iff)
huffman@21164: apply (insert linorder_linear [of "x-a" "b-x"], safe)
huffman@21164: apply (rule_tac x = "x-a" in exI)
huffman@21164: apply (rule_tac [2] x = "b-x" in exI, auto)
huffman@21164: done
huffman@21164: 
huffman@21164: lemma lemma_interval: "[| a < x;  x < b |] ==>
huffman@21164:         \<exists>d::real. 0 < d &  (\<forall>y. \<bar>x-y\<bar> < d --> a \<le> y & y \<le> b)"
huffman@21164: apply (drule lemma_interval_lt, auto)
huffman@21164: apply (auto intro!: exI)
huffman@21164: done
huffman@21164: 
huffman@21164: text{*Rolle's Theorem.
huffman@21164:    If @{term f} is defined and continuous on the closed interval
huffman@21164:    @{text "[a,b]"} and differentiable on the open interval @{text "(a,b)"},
huffman@21164:    and @{term "f(a) = f(b)"},
huffman@21164:    then there exists @{text "x0 \<in> (a,b)"} such that @{term "f'(x0) = 0"}*}
huffman@21164: theorem Rolle:
huffman@21164:   assumes lt: "a < b"
huffman@21164:       and eq: "f(a) = f(b)"
huffman@21164:       and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
huffman@21164:       and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"
huffman@21164:   shows "\<exists>z. a < z & z < b & DERIV f z :> 0"
huffman@21164: proof -
huffman@21164:   have le: "a \<le> b" using lt by simp
huffman@21164:   from isCont_eq_Ub [OF le con]
huffman@21164:   obtain x where x_max: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f z \<le> f x"
huffman@21164:              and alex: "a \<le> x" and xleb: "x \<le> b"
huffman@21164:     by blast
huffman@21164:   from isCont_eq_Lb [OF le con]
huffman@21164:   obtain x' where x'_min: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f x' \<le> f z"
huffman@21164:               and alex': "a \<le> x'" and x'leb: "x' \<le> b"
huffman@21164:     by blast
huffman@21164:   show ?thesis
huffman@21164:   proof cases
huffman@21164:     assume axb: "a < x & x < b"
huffman@21164:         --{*@{term f} attains its maximum within the interval*}
huffman@21164:     hence ax: "a<x" and xb: "x<b" by auto
huffman@21164:     from lemma_interval [OF ax xb]
huffman@21164:     obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
huffman@21164:       by blast
huffman@21164:     hence bound': "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> f y \<le> f x" using x_max
huffman@21164:       by blast
huffman@21164:     from differentiableD [OF dif [OF axb]]
huffman@21164:     obtain l where der: "DERIV f x :> l" ..
huffman@21164:     have "l=0" by (rule DERIV_local_max [OF der d bound'])
huffman@21164:         --{*the derivative at a local maximum is zero*}
huffman@21164:     thus ?thesis using ax xb der by auto
huffman@21164:   next
huffman@21164:     assume notaxb: "~ (a < x & x < b)"
huffman@21164:     hence xeqab: "x=a | x=b" using alex xleb by arith
huffman@21164:     hence fb_eq_fx: "f b = f x" by (auto simp add: eq)
huffman@21164:     show ?thesis
huffman@21164:     proof cases
huffman@21164:       assume ax'b: "a < x' & x' < b"
huffman@21164:         --{*@{term f} attains its minimum within the interval*}
huffman@21164:       hence ax': "a<x'" and x'b: "x'<b" by auto
huffman@21164:       from lemma_interval [OF ax' x'b]
huffman@21164:       obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
huffman@21164:   by blast
huffman@21164:       hence bound': "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> f x' \<le> f y" using x'_min
huffman@21164:   by blast
huffman@21164:       from differentiableD [OF dif [OF ax'b]]
huffman@21164:       obtain l where der: "DERIV f x' :> l" ..
huffman@21164:       have "l=0" by (rule DERIV_local_min [OF der d bound'])
huffman@21164:         --{*the derivative at a local minimum is zero*}
huffman@21164:       thus ?thesis using ax' x'b der by auto
huffman@21164:     next
huffman@21164:       assume notax'b: "~ (a < x' & x' < b)"
huffman@21164:         --{*@{term f} is constant througout the interval*}
huffman@21164:       hence x'eqab: "x'=a | x'=b" using alex' x'leb by arith
huffman@21164:       hence fb_eq_fx': "f b = f x'" by (auto simp add: eq)
huffman@21164:       from dense [OF lt]
huffman@21164:       obtain r where ar: "a < r" and rb: "r < b" by blast
huffman@21164:       from lemma_interval [OF ar rb]
huffman@21164:       obtain d where d: "0<d" and bound: "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
huffman@21164:   by blast
huffman@21164:       have eq_fb: "\<forall>z. a \<le> z --> z \<le> b --> f z = f b"
huffman@21164:       proof (clarify)
huffman@21164:         fix z::real
huffman@21164:         assume az: "a \<le> z" and zb: "z \<le> b"
huffman@21164:         show "f z = f b"
huffman@21164:         proof (rule order_antisym)
huffman@21164:           show "f z \<le> f b" by (simp add: fb_eq_fx x_max az zb)
huffman@21164:           show "f b \<le> f z" by (simp add: fb_eq_fx' x'_min az zb)
huffman@21164:         qed
huffman@21164:       qed
huffman@21164:       have bound': "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> f r = f y"
huffman@21164:       proof (intro strip)
huffman@21164:         fix y::real
huffman@21164:         assume lt: "\<bar>r-y\<bar> < d"
huffman@21164:         hence "f y = f b" by (simp add: eq_fb bound)
huffman@21164:         thus "f r = f y" by (simp add: eq_fb ar rb order_less_imp_le)
huffman@21164:       qed
huffman@21164:       from differentiableD [OF dif [OF conjI [OF ar rb]]]
huffman@21164:       obtain l where der: "DERIV f r :> l" ..
huffman@21164:       have "l=0" by (rule DERIV_local_const [OF der d bound'])
huffman@21164:         --{*the derivative of a constant function is zero*}
huffman@21164:       thus ?thesis using ar rb der by auto
huffman@21164:     qed
huffman@21164:   qed
huffman@21164: qed
huffman@21164: 
huffman@21164: 
huffman@21164: subsection{*Mean Value Theorem*}
huffman@21164: 
huffman@21164: lemma lemma_MVT:
huffman@21164:      "f a - (f b - f a)/(b-a) * a = f b - (f b - f a)/(b-a) * (b::real)"
huffman@21164: proof cases
huffman@21164:   assume "a=b" thus ?thesis by simp
huffman@21164: next
huffman@21164:   assume "a\<noteq>b"
huffman@21164:   hence ba: "b-a \<noteq> 0" by arith
huffman@21164:   show ?thesis
huffman@21164:     by (rule real_mult_left_cancel [OF ba, THEN iffD1],
huffman@21164:         simp add: right_diff_distrib,
huffman@21164:         simp add: left_diff_distrib)
huffman@21164: qed
huffman@21164: 
huffman@21164: theorem MVT:
huffman@21164:   assumes lt:  "a < b"
huffman@21164:       and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
huffman@21164:       and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"
huffman@21164:   shows "\<exists>l z. a < z & z < b & DERIV f z :> l &
huffman@21164:                    (f(b) - f(a) = (b-a) * l)"
huffman@21164: proof -
huffman@21164:   let ?F = "%x. f x - ((f b - f a) / (b-a)) * x"
huffman@21164:   have contF: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?F x" using con
huffman@21164:     by (fast intro: isCont_diff isCont_const isCont_mult isCont_Id)
huffman@21164:   have difF: "\<forall>x. a < x \<and> x < b \<longrightarrow> ?F differentiable x"
huffman@21164:   proof (clarify)
huffman@21164:     fix x::real
huffman@21164:     assume ax: "a < x" and xb: "x < b"
huffman@21164:     from differentiableD [OF dif [OF conjI [OF ax xb]]]
huffman@21164:     obtain l where der: "DERIV f x :> l" ..
huffman@21164:     show "?F differentiable x"
huffman@21164:       by (rule differentiableI [where D = "l - (f b - f a)/(b-a)"],
huffman@21164:           blast intro: DERIV_diff DERIV_cmult_Id der)
huffman@21164:   qed
huffman@21164:   from Rolle [where f = ?F, OF lt lemma_MVT contF difF]
huffman@21164:   obtain z where az: "a < z" and zb: "z < b" and der: "DERIV ?F z :> 0"
huffman@21164:     by blast
huffman@21164:   have "DERIV (%x. ((f b - f a)/(b-a)) * x) z :> (f b - f a)/(b-a)"
huffman@21164:     by (rule DERIV_cmult_Id)
huffman@21164:   hence derF: "DERIV (\<lambda>x. ?F x + (f b - f a) / (b - a) * x) z
huffman@21164:                    :> 0 + (f b - f a) / (b - a)"
huffman@21164:     by (rule DERIV_add [OF der])
huffman@21164:   show ?thesis
huffman@21164:   proof (intro exI conjI)
huffman@21164:     show "a < z" .
huffman@21164:     show "z < b" .
huffman@21164:     show "f b - f a = (b - a) * ((f b - f a)/(b-a))" by (simp)
huffman@21164:     show "DERIV f z :> ((f b - f a)/(b-a))"  using derF by simp
huffman@21164:   qed
huffman@21164: qed
huffman@21164: 
huffman@21164: 
huffman@21164: text{*A function is constant if its derivative is 0 over an interval.*}
huffman@21164: 
huffman@21164: lemma DERIV_isconst_end:
huffman@21164:   fixes f :: "real => real"
huffman@21164:   shows "[| a < b;
huffman@21164:          \<forall>x. a \<le> x & x \<le> b --> isCont f x;
huffman@21164:          \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
huffman@21164:         ==> f b = f a"
huffman@21164: apply (drule MVT, assumption)
huffman@21164: apply (blast intro: differentiableI)
huffman@21164: apply (auto dest!: DERIV_unique simp add: diff_eq_eq)
huffman@21164: done
huffman@21164: 
huffman@21164: lemma DERIV_isconst1:
huffman@21164:   fixes f :: "real => real"
huffman@21164:   shows "[| a < b;
huffman@21164:          \<forall>x. a \<le> x & x \<le> b --> isCont f x;
huffman@21164:          \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
huffman@21164:         ==> \<forall>x. a \<le> x & x \<le> b --> f x = f a"
huffman@21164: apply safe
huffman@21164: apply (drule_tac x = a in order_le_imp_less_or_eq, safe)
huffman@21164: apply (drule_tac b = x in DERIV_isconst_end, auto)
huffman@21164: done
huffman@21164: 
huffman@21164: lemma DERIV_isconst2:
huffman@21164:   fixes f :: "real => real"
huffman@21164:   shows "[| a < b;
huffman@21164:          \<forall>x. a \<le> x & x \<le> b --> isCont f x;
huffman@21164:          \<forall>x. a < x & x < b --> DERIV f x :> 0;
huffman@21164:          a \<le> x; x \<le> b |]
huffman@21164:         ==> f x = f a"
huffman@21164: apply (blast dest: DERIV_isconst1)
huffman@21164: done
huffman@21164: 
huffman@21164: lemma DERIV_isconst_all:
huffman@21164:   fixes f :: "real => real"
huffman@21164:   shows "\<forall>x. DERIV f x :> 0 ==> f(x) = f(y)"
huffman@21164: apply (rule linorder_cases [of x y])
huffman@21164: apply (blast intro: sym DERIV_isCont DERIV_isconst_end)+
huffman@21164: done
huffman@21164: 
huffman@21164: lemma DERIV_const_ratio_const:
huffman@21164:      "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a)) = (b-a) * k"
huffman@21164: apply (rule linorder_cases [of a b], auto)
huffman@21164: apply (drule_tac [!] f = f in MVT)
huffman@21164: apply (auto dest: DERIV_isCont DERIV_unique simp add: differentiable_def)
huffman@21164: apply (auto dest: DERIV_unique simp add: left_distrib diff_minus)
huffman@21164: done
huffman@21164: 
huffman@21164: lemma DERIV_const_ratio_const2:
huffman@21164:      "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a))/(b-a) = k"
huffman@21164: apply (rule_tac c1 = "b-a" in real_mult_right_cancel [THEN iffD1])
huffman@21164: apply (auto dest!: DERIV_const_ratio_const simp add: mult_assoc)
huffman@21164: done
huffman@21164: 
huffman@21164: lemma real_average_minus_first [simp]: "((a + b) /2 - a) = (b-a)/(2::real)"
huffman@21164: by (simp)
huffman@21164: 
huffman@21164: lemma real_average_minus_second [simp]: "((b + a)/2 - a) = (b-a)/(2::real)"
huffman@21164: by (simp)
huffman@21164: 
huffman@21164: text{*Gallileo's "trick": average velocity = av. of end velocities*}
huffman@21164: 
huffman@21164: lemma DERIV_const_average:
huffman@21164:   fixes v :: "real => real"
huffman@21164:   assumes neq: "a \<noteq> (b::real)"
huffman@21164:       and der: "\<forall>x. DERIV v x :> k"
huffman@21164:   shows "v ((a + b)/2) = (v a + v b)/2"
huffman@21164: proof (cases rule: linorder_cases [of a b])
huffman@21164:   case equal with neq show ?thesis by simp
huffman@21164: next
huffman@21164:   case less
huffman@21164:   have "(v b - v a) / (b - a) = k"
huffman@21164:     by (rule DERIV_const_ratio_const2 [OF neq der])
huffman@21164:   hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
huffman@21164:   moreover have "(v ((a + b) / 2) - v a) / ((a + b) / 2 - a) = k"
huffman@21164:     by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
huffman@21164:   ultimately show ?thesis using neq by force
huffman@21164: next
huffman@21164:   case greater
huffman@21164:   have "(v b - v a) / (b - a) = k"
huffman@21164:     by (rule DERIV_const_ratio_const2 [OF neq der])
huffman@21164:   hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
huffman@21164:   moreover have " (v ((b + a) / 2) - v a) / ((b + a) / 2 - a) = k"
huffman@21164:     by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
huffman@21164:   ultimately show ?thesis using neq by (force simp add: add_commute)
huffman@21164: qed
huffman@21164: 
huffman@21164: 
huffman@21164: text{*Dull lemma: an continuous injection on an interval must have a
huffman@21164: strict maximum at an end point, not in the middle.*}
huffman@21164: 
huffman@21164: lemma lemma_isCont_inj:
huffman@21164:   fixes f :: "real \<Rightarrow> real"
huffman@21164:   assumes d: "0 < d"
huffman@21164:       and inj [rule_format]: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
huffman@21164:       and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
huffman@21164:   shows "\<exists>z. \<bar>z-x\<bar> \<le> d & f x < f z"
huffman@21164: proof (rule ccontr)
huffman@21164:   assume  "~ (\<exists>z. \<bar>z-x\<bar> \<le> d & f x < f z)"
huffman@21164:   hence all [rule_format]: "\<forall>z. \<bar>z - x\<bar> \<le> d --> f z \<le> f x" by auto
huffman@21164:   show False
huffman@21164:   proof (cases rule: linorder_le_cases [of "f(x-d)" "f(x+d)"])
huffman@21164:     case le
huffman@21164:     from d cont all [of "x+d"]
huffman@21164:     have flef: "f(x+d) \<le> f x"
huffman@21164:      and xlex: "x - d \<le> x"
huffman@21164:      and cont': "\<forall>z. x - d \<le> z \<and> z \<le> x \<longrightarrow> isCont f z"
huffman@21164:        by (auto simp add: abs_if)
huffman@21164:     from IVT [OF le flef xlex cont']
huffman@21164:     obtain x' where "x-d \<le> x'" "x' \<le> x" "f x' = f(x+d)" by blast
huffman@21164:     moreover
huffman@21164:     hence "g(f x') = g (f(x+d))" by simp
huffman@21164:     ultimately show False using d inj [of x'] inj [of "x+d"]
huffman@21164:       by (simp add: abs_le_interval_iff)
huffman@21164:   next
huffman@21164:     case ge
huffman@21164:     from d cont all [of "x-d"]
huffman@21164:     have flef: "f(x-d) \<le> f x"
huffman@21164:      and xlex: "x \<le> x+d"
huffman@21164:      and cont': "\<forall>z. x \<le> z \<and> z \<le> x+d \<longrightarrow> isCont f z"
huffman@21164:        by (auto simp add: abs_if)
huffman@21164:     from IVT2 [OF ge flef xlex cont']
huffman@21164:     obtain x' where "x \<le> x'" "x' \<le> x+d" "f x' = f(x-d)" by blast
huffman@21164:     moreover
huffman@21164:     hence "g(f x') = g (f(x-d))" by simp
huffman@21164:     ultimately show False using d inj [of x'] inj [of "x-d"]
huffman@21164:       by (simp add: abs_le_interval_iff)
huffman@21164:   qed
huffman@21164: qed
huffman@21164: 
huffman@21164: 
huffman@21164: text{*Similar version for lower bound.*}
huffman@21164: 
huffman@21164: lemma lemma_isCont_inj2:
huffman@21164:   fixes f g :: "real \<Rightarrow> real"
huffman@21164:   shows "[|0 < d; \<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z;
huffman@21164:         \<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z |]
huffman@21164:       ==> \<exists>z. \<bar>z-x\<bar> \<le> d & f z < f x"
huffman@21164: apply (insert lemma_isCont_inj
huffman@21164:           [where f = "%x. - f x" and g = "%y. g(-y)" and x = x and d = d])
huffman@21164: apply (simp add: isCont_minus linorder_not_le)
huffman@21164: done
huffman@21164: 
huffman@21164: text{*Show there's an interval surrounding @{term "f(x)"} in
huffman@21164: @{text "f[[x - d, x + d]]"} .*}
huffman@21164: 
huffman@21164: lemma isCont_inj_range:
huffman@21164:   fixes f :: "real \<Rightarrow> real"
huffman@21164:   assumes d: "0 < d"
huffman@21164:       and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
huffman@21164:       and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
huffman@21164:   shows "\<exists>e>0. \<forall>y. \<bar>y - f x\<bar> \<le> e --> (\<exists>z. \<bar>z-x\<bar> \<le> d & f z = y)"
huffman@21164: proof -
huffman@21164:   have "x-d \<le> x+d" "\<forall>z. x-d \<le> z \<and> z \<le> x+d \<longrightarrow> isCont f z" using cont d
huffman@21164:     by (auto simp add: abs_le_interval_iff)
huffman@21164:   from isCont_Lb_Ub [OF this]
huffman@21164:   obtain L M
huffman@21164:   where all1 [rule_format]: "\<forall>z. x-d \<le> z \<and> z \<le> x+d \<longrightarrow> L \<le> f z \<and> f z \<le> M"
huffman@21164:     and all2 [rule_format]:
huffman@21164:            "\<forall>y. L \<le> y \<and> y \<le> M \<longrightarrow> (\<exists>z. x-d \<le> z \<and> z \<le> x+d \<and> f z = y)"
huffman@21164:     by auto
huffman@21164:   with d have "L \<le> f x & f x \<le> M" by simp
huffman@21164:   moreover have "L \<noteq> f x"
huffman@21164:   proof -
huffman@21164:     from lemma_isCont_inj2 [OF d inj cont]
huffman@21164:     obtain u where "\<bar>u - x\<bar> \<le> d" "f u < f x"  by auto
huffman@21164:     thus ?thesis using all1 [of u] by arith
huffman@21164:   qed
huffman@21164:   moreover have "f x \<noteq> M"
huffman@21164:   proof -
huffman@21164:     from lemma_isCont_inj [OF d inj cont]
huffman@21164:     obtain u where "\<bar>u - x\<bar> \<le> d" "f x < f u"  by auto
huffman@21164:     thus ?thesis using all1 [of u] by arith
huffman@21164:   qed
huffman@21164:   ultimately have "L < f x & f x < M" by arith
huffman@21164:   hence "0 < f x - L" "0 < M - f x" by arith+
huffman@21164:   from real_lbound_gt_zero [OF this]
huffman@21164:   obtain e where e: "0 < e" "e < f x - L" "e < M - f x" by auto
huffman@21164:   thus ?thesis
huffman@21164:   proof (intro exI conjI)
huffman@21164:     show "0<e" .
huffman@21164:     show "\<forall>y. \<bar>y - f x\<bar> \<le> e \<longrightarrow> (\<exists>z. \<bar>z - x\<bar> \<le> d \<and> f z = y)"
huffman@21164:     proof (intro strip)
huffman@21164:       fix y::real
huffman@21164:       assume "\<bar>y - f x\<bar> \<le> e"
huffman@21164:       with e have "L \<le> y \<and> y \<le> M" by arith
huffman@21164:       from all2 [OF this]
huffman@21164:       obtain z where "x - d \<le> z" "z \<le> x + d" "f z = y" by blast
huffman@21164:       thus "\<exists>z. \<bar>z - x\<bar> \<le> d \<and> f z = y"
huffman@21164:         by (force simp add: abs_le_interval_iff)
huffman@21164:     qed
huffman@21164:   qed
huffman@21164: qed
huffman@21164: 
huffman@21164: 
huffman@21164: text{*Continuity of inverse function*}
huffman@21164: 
huffman@21164: lemma isCont_inverse_function:
huffman@21164:   fixes f g :: "real \<Rightarrow> real"
huffman@21164:   assumes d: "0 < d"
huffman@21164:       and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
huffman@21164:       and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
huffman@21164:   shows "isCont g (f x)"
huffman@21164: proof (simp add: isCont_iff LIM_eq)
huffman@21164:   show "\<forall>r. 0 < r \<longrightarrow>
huffman@21164:          (\<exists>s>0. \<forall>z. z\<noteq>0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>g(f x + z) - g(f x)\<bar> < r)"
huffman@21164:   proof (intro strip)
huffman@21164:     fix r::real
huffman@21164:     assume r: "0<r"
huffman@21164:     from real_lbound_gt_zero [OF r d]
huffman@21164:     obtain e where e: "0 < e" and e_lt: "e < r \<and> e < d" by blast
huffman@21164:     with inj cont
huffman@21164:     have e_simps: "\<forall>z. \<bar>z-x\<bar> \<le> e --> g (f z) = z"
huffman@21164:                   "\<forall>z. \<bar>z-x\<bar> \<le> e --> isCont f z"   by auto
huffman@21164:     from isCont_inj_range [OF e this]
huffman@21164:     obtain e' where e': "0 < e'"
huffman@21164:         and all: "\<forall>y. \<bar>y - f x\<bar> \<le> e' \<longrightarrow> (\<exists>z. \<bar>z - x\<bar> \<le> e \<and> f z = y)"
huffman@21164:           by blast
huffman@21164:     show "\<exists>s>0. \<forall>z. z\<noteq>0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>g(f x + z) - g(f x)\<bar> < r"
huffman@21164:     proof (intro exI conjI)
huffman@21164:       show "0<e'" .
huffman@21164:       show "\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < e' \<longrightarrow> \<bar>g (f x + z) - g (f x)\<bar> < r"
huffman@21164:       proof (intro strip)
huffman@21164:         fix z::real
huffman@21164:         assume z: "z \<noteq> 0 \<and> \<bar>z\<bar> < e'"
huffman@21164:         with e e_lt e_simps all [rule_format, of "f x + z"]
huffman@21164:         show "\<bar>g (f x + z) - g (f x)\<bar> < r" by force
huffman@21164:       qed
huffman@21164:     qed
huffman@21164:   qed
huffman@21164: qed
huffman@21164: 
huffman@21164: theorem GMVT:
huffman@21164:   assumes alb: "a < b"
huffman@21164:   and fc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
huffman@21164:   and fd: "\<forall>x. a < x \<and> x < b \<longrightarrow> f differentiable x"
huffman@21164:   and gc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont g x"
huffman@21164:   and gd: "\<forall>x. a < x \<and> x < b \<longrightarrow> g differentiable x"
huffman@21164:   shows "\<exists>g'c f'c c. DERIV g c :> g'c \<and> DERIV f c :> f'c \<and> a < c \<and> c < b \<and> ((f b - f a) * g'c) = ((g b - g a) * f'c)"
huffman@21164: proof -
huffman@21164:   let ?h = "\<lambda>x. (f b - f a)*(g x) - (g b - g a)*(f x)"
huffman@21164:   from prems have "a < b" by simp
huffman@21164:   moreover have "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?h x"
huffman@21164:   proof -
huffman@21164:     have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. f b - f a) x" by simp
huffman@21164:     with gc have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. (f b - f a) * g x) x"
huffman@21164:       by (auto intro: isCont_mult)
huffman@21164:     moreover
huffman@21164:     have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. g b - g a) x" by simp
huffman@21164:     with fc have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. (g b - g a) * f x) x"
huffman@21164:       by (auto intro: isCont_mult)
huffman@21164:     ultimately show ?thesis
huffman@21164:       by (fastsimp intro: isCont_diff)
huffman@21164:   qed
huffman@21164:   moreover
huffman@21164:   have "\<forall>x. a < x \<and> x < b \<longrightarrow> ?h differentiable x"
huffman@21164:   proof -
huffman@21164:     have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. f b - f a) differentiable x" by (simp add: differentiable_const)
huffman@21164:     with gd have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. (f b - f a) * g x) differentiable x" by (simp add: differentiable_mult)
huffman@21164:     moreover
huffman@21164:     have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. g b - g a) differentiable x" by (simp add: differentiable_const)
huffman@21164:     with fd have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. (g b - g a) * f x) differentiable x" by (simp add: differentiable_mult)
huffman@21164:     ultimately show ?thesis by (simp add: differentiable_diff)
huffman@21164:   qed
huffman@21164:   ultimately have "\<exists>l z. a < z \<and> z < b \<and> DERIV ?h z :> l \<and> ?h b - ?h a = (b - a) * l" by (rule MVT)
huffman@21164:   then obtain l where ldef: "\<exists>z. a < z \<and> z < b \<and> DERIV ?h z :> l \<and> ?h b - ?h a = (b - a) * l" ..
huffman@21164:   then obtain c where cdef: "a < c \<and> c < b \<and> DERIV ?h c :> l \<and> ?h b - ?h a = (b - a) * l" ..
huffman@21164: 
huffman@21164:   from cdef have cint: "a < c \<and> c < b" by auto
huffman@21164:   with gd have "g differentiable c" by simp
huffman@21164:   hence "\<exists>D. DERIV g c :> D" by (rule differentiableD)
huffman@21164:   then obtain g'c where g'cdef: "DERIV g c :> g'c" ..
huffman@21164: 
huffman@21164:   from cdef have "a < c \<and> c < b" by auto
huffman@21164:   with fd have "f differentiable c" by simp
huffman@21164:   hence "\<exists>D. DERIV f c :> D" by (rule differentiableD)
huffman@21164:   then obtain f'c where f'cdef: "DERIV f c :> f'c" ..
huffman@21164: 
huffman@21164:   from cdef have "DERIV ?h c :> l" by auto
huffman@21164:   moreover
huffman@21164:   {
huffman@21164:     from g'cdef have "DERIV (\<lambda>x. (f b - f a) * g x) c :> g'c * (f b - f a)"
huffman@21164:       apply (insert DERIV_const [where k="f b - f a"])
huffman@21164:       apply (drule meta_spec [of _ c])
huffman@21164:       apply (drule DERIV_mult [where f="(\<lambda>x. f b - f a)" and g=g])
huffman@21164:       by simp_all
huffman@21164:     moreover from f'cdef have "DERIV (\<lambda>x. (g b - g a) * f x) c :> f'c * (g b - g a)"
huffman@21164:       apply (insert DERIV_const [where k="g b - g a"])
huffman@21164:       apply (drule meta_spec [of _ c])
huffman@21164:       apply (drule DERIV_mult [where f="(\<lambda>x. g b - g a)" and g=f])
huffman@21164:       by simp_all
huffman@21164:     ultimately have "DERIV ?h c :>  g'c * (f b - f a) - f'c * (g b - g a)"
huffman@21164:       by (simp add: DERIV_diff)
huffman@21164:   }
huffman@21164:   ultimately have leq: "l =  g'c * (f b - f a) - f'c * (g b - g a)" by (rule DERIV_unique)
huffman@21164: 
huffman@21164:   {
huffman@21164:     from cdef have "?h b - ?h a = (b - a) * l" by auto
huffman@21164:     also with leq have "\<dots> = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
huffman@21164:     finally have "?h b - ?h a = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
huffman@21164:   }
huffman@21164:   moreover
huffman@21164:   {
huffman@21164:     have "?h b - ?h a =
huffman@21164:          ((f b)*(g b) - (f a)*(g b) - (g b)*(f b) + (g a)*(f b)) -
huffman@21164:           ((f b)*(g a) - (f a)*(g a) - (g b)*(f a) + (g a)*(f a))"
huffman@21164:       by (simp add: mult_ac add_ac real_diff_mult_distrib)
huffman@21164:     hence "?h b - ?h a = 0" by auto
huffman@21164:   }
huffman@21164:   ultimately have "(b - a) * (g'c * (f b - f a) - f'c * (g b - g a)) = 0" by auto
huffman@21164:   with alb have "g'c * (f b - f a) - f'c * (g b - g a) = 0" by simp
huffman@21164:   hence "g'c * (f b - f a) = f'c * (g b - g a)" by simp
huffman@21164:   hence "(f b - f a) * g'c = (g b - g a) * f'c" by (simp add: mult_ac)
huffman@21164: 
huffman@21164:   with g'cdef f'cdef cint show ?thesis by auto
huffman@21164: qed
huffman@21164: 
huffman@21164: end




