--- a/src/HOL/Hyperreal/Lim.thy Sat Nov 04 00:11:11 2006 +0100
+++ b/src/HOL/Hyperreal/Lim.thy Sat Nov 04 00:12:06 2006 +0100
@@ -3,10 +3,9 @@
Author : Jacques D. Fleuriot
Copyright : 1998 University of Cambridge
Conversion to Isar and new proofs by Lawrence C Paulson, 2004
- GMVT by Benjamin Porter, 2005
*)
-header{*Limits, Continuity and Differentiation*}
+header{* Limits and Continuity *}
theory Lim
imports SEQ
@@ -34,28 +33,6 @@
"isNSCont f a = (\<forall>y. y @= star_of a -->
( *f* f) y @= star_of (f a))"
- deriv :: "[real \<Rightarrow> 'a::real_normed_vector, real, 'a] \<Rightarrow> bool"
- --{*Differentiation: D is derivative of function f at x*}
- ("(DERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60)
- "DERIV f x :> D = ((%h. (f(x + h) - f x) /# h) -- 0 --> D)"
-
- nsderiv :: "[real=>real,real,real] => bool"
- ("(NSDERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60)
- "NSDERIV f x :> D = (\<forall>h \<in> Infinitesimal - {0}.
- (( *f* f)(hypreal_of_real x + h)
- - hypreal_of_real (f x))/h @= hypreal_of_real D)"
-
- differentiable :: "[real=>real,real] => bool" (infixl "differentiable" 60)
- "f differentiable x = (\<exists>D. DERIV f x :> D)"
-
- NSdifferentiable :: "[real=>real,real] => bool"
- (infixl "NSdifferentiable" 60)
- "f NSdifferentiable x = (\<exists>D. NSDERIV f x :> D)"
-
- increment :: "[real=>real,real,hypreal] => hypreal"
- "increment f x h = (@inc. f NSdifferentiable x &
- inc = ( *f* f)(hypreal_of_real x + h) - hypreal_of_real (f x))"
-
isUCont :: "['a::real_normed_vector => 'b::real_normed_vector] => bool"
"isUCont f = (\<forall>r>0. \<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r)"
@@ -63,16 +40,6 @@
"isNSUCont f = (\<forall>x y. x @= y --> ( *f* f) x @= ( *f* f) y)"
-consts
- Bolzano_bisect :: "[real*real=>bool, real, real, nat] => (real*real)"
-primrec
- "Bolzano_bisect P a b 0 = (a,b)"
- "Bolzano_bisect P a b (Suc n) =
- (let (x,y) = Bolzano_bisect P a b n
- in if P(x, (x+y)/2) then ((x+y)/2, y)
- else (x, (x+y)/2))"
-
-
subsection {* Limits of Functions *}
subsubsection {* Purely standard proofs *}
@@ -92,7 +59,7 @@
==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r"
by (simp add: LIM_eq)
-lemma LIM_shift: "f -- a --> L \<Longrightarrow> (\<lambda>x. f (x + k)) -- a - k --> L"
+lemma LIM_offset: "f -- a --> L \<Longrightarrow> (\<lambda>x. f (x + k)) -- a - k --> L"
apply (rule LIM_I)
apply (drule_tac r="r" in LIM_D, safe)
apply (rule_tac x="s" in exI, safe)
@@ -680,249 +647,9 @@
by transfer
qed
-subsection {* Derivatives *}
-
-subsubsection {* Purely standard proofs *}
-
-lemma DERIV_iff: "(DERIV f x :> D) = ((%h. (f(x + h) - f(x))/#h) -- 0 --> D)"
-by (simp add: deriv_def)
-
-lemma DERIV_D: "DERIV f x :> D ==> (%h. (f(x + h) - f(x))/#h) -- 0 --> D"
-by (simp add: deriv_def)
-
-lemma DERIV_const [simp]: "DERIV (\<lambda>x. k) x :> 0"
-by (simp add: deriv_def)
-
-lemma DERIV_Id [simp]: "DERIV (\<lambda>x. x) x :> 1"
-by (simp add: deriv_def real_scaleR_def cong: LIM_cong)
-
-lemma add_diff_add:
- fixes a b c d :: "'a::ab_group_add"
- shows "(a + c) - (b + d) = (a - b) + (c - d)"
-by simp
-
-lemma DERIV_add:
- "\<lbrakk>DERIV f x :> D; DERIV g x :> E\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. f x + g x) x :> D + E"
-by (simp only: deriv_def add_diff_add scaleR_right_distrib LIM_add)
-
-lemma DERIV_minus:
- "DERIV f x :> D \<Longrightarrow> DERIV (\<lambda>x. - f x) x :> - D"
-by (simp only: deriv_def minus_diff_minus scaleR_minus_right LIM_minus)
-
-lemma DERIV_diff:
- "\<lbrakk>DERIV f x :> D; DERIV g x :> E\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. f x - g x) x :> D - E"
-by (simp only: diff_def DERIV_add DERIV_minus)
-
-lemma DERIV_add_minus:
- "\<lbrakk>DERIV f x :> D; DERIV g x :> E\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. f x + - g x) x :> D + - E"
-by (simp only: DERIV_add DERIV_minus)
-
-lemma DERIV_isCont: "DERIV f x :> D \<Longrightarrow> isCont f x"
-proof (unfold isCont_iff)
- assume "DERIV f x :> D"
- hence "(\<lambda>h. (f(x+h) - f(x)) /# h) -- 0 --> D"
- by (rule DERIV_D)
- hence "(\<lambda>h. h *# ((f(x+h) - f(x)) /# h)) -- 0 --> 0 *# D"
- by (intro LIM_scaleR LIM_self)
- hence "(\<lambda>h. (f(x+h) - f(x))) -- 0 --> 0"
- by (simp cong: LIM_cong)
- thus "(\<lambda>h. f(x+h)) -- 0 --> f(x)"
- by (simp add: LIM_def)
-qed
-
-lemma DERIV_mult_lemma:
- fixes a b c d :: "'a::real_algebra"
- shows "(a * b - c * d) /# h = a * ((b - d) /# h) + ((a - c) /# h) * d"
-by (simp add: diff_minus scaleR_right_distrib [symmetric] ring_distrib)
-
-lemma DERIV_mult':
- fixes f g :: "real \<Rightarrow> 'a::real_normed_algebra"
- assumes f: "DERIV f x :> D"
- assumes g: "DERIV g x :> E"
- shows "DERIV (\<lambda>x. f x * g x) x :> f x * E + D * g x"
-proof (unfold deriv_def)
- from f have "isCont f x"
- by (rule DERIV_isCont)
- hence "(\<lambda>h. f(x+h)) -- 0 --> f x"
- by (simp only: isCont_iff)
- hence "(\<lambda>h. f(x+h) * ((g(x+h) - g x) /# h) +
- ((f(x+h) - f x) /# h) * g x)
- -- 0 --> f x * E + D * g x"
- by (intro LIM_add LIM_mult2 LIM_const DERIV_D f g)
- thus "(\<lambda>h. (f(x+h) * g(x+h) - f x * g x) /# h)
- -- 0 --> f x * E + D * g x"
- by (simp only: DERIV_mult_lemma)
-qed
-
-lemma DERIV_mult:
- fixes f g :: "real \<Rightarrow> 'a::{real_normed_algebra,comm_ring}" shows
- "[| DERIV f x :> Da; DERIV g x :> Db |]
- ==> DERIV (%x. f x * g x) x :> (Da * g(x)) + (Db * f(x))"
-by (drule (1) DERIV_mult', simp only: mult_commute add_commute)
-
-lemma DERIV_unique:
- "[| DERIV f x :> D; DERIV f x :> E |] ==> D = E"
-apply (simp add: deriv_def)
-apply (blast intro: LIM_unique)
-done
-
-text{*Differentiation of finite sum*}
-
-lemma DERIV_sumr [rule_format (no_asm)]:
- "(\<forall>r. m \<le> r & r < (m + n) --> DERIV (%x. f r x) x :> (f' r x))
- --> DERIV (%x. \<Sum>n=m..<n::nat. f n x :: real) x :> (\<Sum>r=m..<n. f' r x)"
-apply (induct "n")
-apply (auto intro: DERIV_add)
-done
-
-text{*Alternative definition for differentiability*}
-
-lemma DERIV_LIM_iff:
- "((%h::real. (f(a + h) - f(a)) / h) -- 0 --> D) =
- ((%x. (f(x)-f(a)) / (x-a)) -- a --> D)"
-apply (rule iffI)
-apply (drule_tac k="- a" in LIM_shift)
-apply (simp add: diff_minus)
-apply (drule_tac k="a" in LIM_shift)
-apply (simp add: add_commute)
-done
-
-lemma DERIV_LIM_iff':
- "((%h::real. (f(a + h) - f(a)) /# h) -- 0 --> D) =
- ((%x. (f(x)-f(a)) /# (x-a)) -- a --> D)"
-apply (rule iffI)
-apply (drule_tac k="- a" in LIM_shift)
-apply (simp add: diff_minus)
-apply (drule_tac k="a" in LIM_shift)
-apply (simp add: add_commute)
-done
-
-lemma DERIV_iff2: "(DERIV f x :> D) = ((%z. (f(z) - f(x)) /# (z-x)) -- x --> D)"
-by (simp add: deriv_def diff_minus [symmetric] DERIV_LIM_iff')
-
-lemma inverse_diff_inverse:
- "\<lbrakk>(a::'a::division_ring) \<noteq> 0; b \<noteq> 0\<rbrakk>
- \<Longrightarrow> inverse a - inverse b = - (inverse a * (a - b) * inverse b)"
-by (simp add: right_diff_distrib left_diff_distrib mult_assoc)
-
-lemma DERIV_inverse_lemma:
- "\<lbrakk>a \<noteq> 0; b \<noteq> (0::'a::real_normed_div_algebra)\<rbrakk>
- \<Longrightarrow> (inverse a - inverse b) /# h
- = - (inverse a * ((a - b) /# h) * inverse b)"
-by (simp add: inverse_diff_inverse)
-
-lemma LIM_equal2:
- assumes 1: "0 < R"
- assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < R\<rbrakk> \<Longrightarrow> f x = g x"
- shows "g -- a --> l \<Longrightarrow> f -- a --> l"
-apply (unfold LIM_def, safe)
-apply (drule_tac x="r" in spec, safe)
-apply (rule_tac x="min s R" in exI, safe)
-apply (simp add: 1)
-apply (simp add: 2)
-done
-
-lemma DERIV_inverse':
- fixes f :: "real \<Rightarrow> 'a::real_normed_div_algebra"
- assumes der: "DERIV f x :> D"
- assumes neq: "f x \<noteq> 0"
- shows "DERIV (\<lambda>x. inverse (f x)) x :> - (inverse (f x) * D * inverse (f x))"
- (is "DERIV _ _ :> ?E")
-proof (unfold DERIV_iff2)
- from der have lim_f: "f -- x --> f x"
- by (rule DERIV_isCont [unfolded isCont_def])
-
- from neq have "0 < norm (f x)" by simp
- with LIM_D [OF lim_f] obtain s
- where s: "0 < s"
- and less_fx: "\<And>z. \<lbrakk>z \<noteq> x; norm (z - x) < s\<rbrakk>
- \<Longrightarrow> norm (f z - f x) < norm (f x)"
- by fast
-
- show "(\<lambda>z. (inverse (f z) - inverse (f x)) /# (z - x)) -- x --> ?E"
- proof (rule LIM_equal2 [OF s])
- fix z :: real
- assume "z \<noteq> x" "norm (z - x) < s"
- hence "norm (f z - f x) < norm (f x)" by (rule less_fx)
- hence "f z \<noteq> 0" by auto
- thus "(inverse (f z) - inverse (f x)) /# (z - x) =
- - (inverse (f z) * ((f z - f x) /# (z - x)) * inverse (f x))"
- using neq by (rule DERIV_inverse_lemma)
- next
- from der have "(\<lambda>z. (f z - f x) /# (z - x)) -- x --> D"
- by (unfold DERIV_iff2)
- thus "(\<lambda>z. - (inverse (f z) * ((f z - f x) /# (z - x)) * inverse (f x)))
- -- x --> ?E"
- by (intro LIM_mult2 LIM_inverse LIM_minus LIM_const lim_f neq)
- qed
-qed
-
-lemma DERIV_divide:
- fixes D E :: "'a::{real_normed_div_algebra,field}"
- shows "\<lbrakk>DERIV f x :> D; DERIV g x :> E; g x \<noteq> 0\<rbrakk>
- \<Longrightarrow> DERIV (\<lambda>x. f x / g x) x :> (D * g x - f x * E) / (g x * g x)"
-apply (subgoal_tac "f x * - (inverse (g x) * E * inverse (g x)) +
- D * inverse (g x) = (D * g x - f x * E) / (g x * g x)")
-apply (erule subst)
-apply (unfold divide_inverse)
-apply (erule DERIV_mult')
-apply (erule (1) DERIV_inverse')
-apply (simp add: left_diff_distrib nonzero_inverse_mult_distrib)
-apply (simp add: mult_ac)
-done
-
-lemma DERIV_power_Suc:
- fixes f :: "real \<Rightarrow> 'a::{real_normed_algebra,recpower}"
- assumes f: "DERIV f x :> D"
- shows "DERIV (\<lambda>x. f x ^ Suc n) x :> (of_nat n + 1) *# (D * f x ^ n)"
-proof (induct n)
-case 0
- show ?case by (simp add: power_Suc f)
-case (Suc k)
- from DERIV_mult' [OF f Suc] show ?case
- apply (simp only: of_nat_Suc scaleR_left_distrib scaleR_one)
- apply (simp only: power_Suc right_distrib mult_scaleR_right mult_ac)
- done
-qed
-
-lemma DERIV_power:
- fixes f :: "real \<Rightarrow> 'a::{real_normed_algebra,recpower}"
- assumes f: "DERIV f x :> D"
- shows "DERIV (\<lambda>x. f x ^ n) x :> of_nat n *# (D * f x ^ (n - Suc 0))"
-by (cases "n", simp, simp add: DERIV_power_Suc f)
-
-
-(* ------------------------------------------------------------------------ *)
-(* Caratheodory formulation of derivative at a point: standard proof *)
-(* ------------------------------------------------------------------------ *)
-
-lemma CARAT_DERIV:
- "(DERIV f x :> l) =
- (\<exists>g. (\<forall>z. f z - f x = (z-x) *# g z) & isCont g x & g x = l)"
- (is "?lhs = ?rhs")
-proof
- assume der: "DERIV f x :> l"
- show "\<exists>g. (\<forall>z. f z - f x = (z-x) *# g z) \<and> isCont g x \<and> g x = l"
- proof (intro exI conjI)
- let ?g = "(%z. if z = x then l else (f z - f x) /# (z-x))"
- show "\<forall>z. f z - f x = (z-x) *# ?g z" by (simp)
- show "isCont ?g x" using der
- by (simp add: isCont_iff DERIV_iff diff_minus
- cong: LIM_equal [rule_format])
- show "?g x = l" by simp
- qed
-next
- assume "?rhs"
- then obtain g where
- "(\<forall>z. f z - f x = (z-x) *# g z)" and "isCont g x" and "g x = l" by blast
- thus "(DERIV f x :> l)"
- by (auto simp add: isCont_iff DERIV_iff diff_minus
- cong: LIM_equal [rule_format])
-qed
-
lemma LIM_compose:
+ assumes g: "isCont g l"
assumes f: "f -- a --> l"
- assumes g: "isCont g l"
shows "(\<lambda>x. g (f x)) -- a --> g l"
proof (rule LIM_I)
fix r::real assume r: "0 < r"
@@ -944,1510 +671,28 @@
qed
qed
-lemma DERIV_chain':
- assumes f: "DERIV f x :> D"
- assumes g: "DERIV g (f x) :> E"
- shows "DERIV (\<lambda>x. g (f x)) x :> D *# E"
-proof (unfold DERIV_iff2)
- obtain d where d: "\<forall>y. g y - g (f x) = (y - f x) *# d y"
- and cont_d: "isCont d (f x)" and dfx: "d (f x) = E"
- using CARAT_DERIV [THEN iffD1, OF g] by fast
- from f have "f -- x --> f x"
- by (rule DERIV_isCont [unfolded isCont_def])
- hence "(\<lambda>z. d (f z)) -- x --> d (f x)"
- using cont_d by (rule LIM_compose)
- hence "(\<lambda>z. ((f z - f x) /# (z - x)) *# d (f z))
- -- x --> D *# d (f x)"
- by (rule LIM_scaleR [OF f [unfolded DERIV_iff2]])
- thus "(\<lambda>z. (g (f z) - g (f x)) /# (z - x)) -- x --> D *# E"
- by (simp add: d dfx real_scaleR_def)
-qed
-
-
-subsubsection {* Nonstandard proofs *}
-
-lemma DERIV_NS_iff:
- "(DERIV f x :> D) = ((%h. (f(x + h) - f(x))/#h) -- 0 --NS> D)"
-by (simp add: deriv_def LIM_NSLIM_iff)
-
-lemma NS_DERIV_D: "DERIV f x :> D ==> (%h. (f(x + h) - f(x))/#h) -- 0 --NS> D"
-by (simp add: deriv_def LIM_NSLIM_iff)
-
-lemma NSDeriv_unique:
- "[| NSDERIV f x :> D; NSDERIV f x :> E |] ==> D = E"
-apply (simp add: nsderiv_def)
-apply (cut_tac Infinitesimal_epsilon hypreal_epsilon_not_zero)
-apply (auto dest!: bspec [where x=epsilon]
- intro!: inj_hypreal_of_real [THEN injD]
- dest: approx_trans3)
-done
-
-text {*First NSDERIV in terms of NSLIM*}
-
-text{*first equivalence *}
-lemma NSDERIV_NSLIM_iff:
- "(NSDERIV f x :> D) = ((%h. (f(x + h) - f(x))/h) -- 0 --NS> D)"
-apply (simp add: nsderiv_def NSLIM_def, auto)
-apply (drule_tac x = xa in bspec)
-apply (rule_tac [3] ccontr)
-apply (drule_tac [3] x = h in spec)
-apply (auto simp add: mem_infmal_iff starfun_lambda_cancel)
-done
-
-text{*second equivalence *}
-lemma NSDERIV_NSLIM_iff2:
- "(NSDERIV f x :> D) = ((%z. (f(z) - f(x)) / (z-x)) -- x --NS> D)"
-by (simp add: NSDERIV_NSLIM_iff DERIV_LIM_iff diff_minus [symmetric]
- LIM_NSLIM_iff [symmetric])
-
-(* while we're at it! *)
-lemma NSDERIV_iff2:
- "(NSDERIV f x :> D) =
- (\<forall>w.
- w \<noteq> hypreal_of_real x & w \<approx> hypreal_of_real x -->
- ( *f* (%z. (f z - f x) / (z-x))) w \<approx> hypreal_of_real D)"
-by (simp add: NSDERIV_NSLIM_iff2 NSLIM_def)
-
-(*FIXME DELETE*)
-lemma hypreal_not_eq_minus_iff: "(x \<noteq> a) = (x - a \<noteq> (0::hypreal))"
-by (auto dest: hypreal_eq_minus_iff [THEN iffD2])
-
-lemma NSDERIVD5:
- "(NSDERIV f x :> D) ==>
- (\<forall>u. u \<approx> hypreal_of_real x -->
- ( *f* (%z. f z - f x)) u \<approx> hypreal_of_real D * (u - hypreal_of_real x))"
-apply (auto simp add: NSDERIV_iff2)
-apply (case_tac "u = hypreal_of_real x", auto)
-apply (drule_tac x = u in spec, auto)
-apply (drule_tac c = "u - hypreal_of_real x" and b = "hypreal_of_real D" in approx_mult1)
-apply (drule_tac [!] hypreal_not_eq_minus_iff [THEN iffD1])
-apply (subgoal_tac [2] "( *f* (%z. z-x)) u \<noteq> (0::hypreal) ")
-apply (auto simp add:
- approx_minus_iff [THEN iffD1, THEN mem_infmal_iff [THEN iffD2]]
- Infinitesimal_subset_HFinite [THEN subsetD])
-done
-
-lemma NSDERIVD4:
- "(NSDERIV f x :> D) ==>
- (\<forall>h \<in> Infinitesimal.
- (( *f* f)(hypreal_of_real x + h) -
- hypreal_of_real (f x))\<approx> (hypreal_of_real D) * h)"
-apply (auto simp add: nsderiv_def)
-apply (case_tac "h = (0::hypreal) ")
-apply (auto simp add: diff_minus)
-apply (drule_tac x = h in bspec)
-apply (drule_tac [2] c = h in approx_mult1)
-apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
- simp add: diff_minus)
-done
-
-lemma NSDERIVD3:
- "(NSDERIV f x :> D) ==>
- (\<forall>h \<in> Infinitesimal - {0}.
- (( *f* f)(hypreal_of_real x + h) -
- hypreal_of_real (f x))\<approx> (hypreal_of_real D) * h)"
-apply (auto simp add: nsderiv_def)
-apply (rule ccontr, drule_tac x = h in bspec)
-apply (drule_tac [2] c = h in approx_mult1)
-apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
- simp add: mult_assoc diff_minus)
-done
-
-text{*Differentiability implies continuity
- nice and simple "algebraic" proof*}
-lemma NSDERIV_isNSCont: "NSDERIV f x :> D ==> isNSCont f x"
-apply (auto simp add: nsderiv_def isNSCont_NSLIM_iff NSLIM_def)
-apply (drule approx_minus_iff [THEN iffD1])
-apply (drule hypreal_not_eq_minus_iff [THEN iffD1])
-apply (drule_tac x = "xa - hypreal_of_real x" in bspec)
- prefer 2 apply (simp add: add_assoc [symmetric])
-apply (auto simp add: mem_infmal_iff [symmetric] add_commute)
-apply (drule_tac c = "xa - hypreal_of_real x" in approx_mult1)
-apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
- simp add: mult_assoc)
-apply (drule_tac x3=D in
- HFinite_hypreal_of_real [THEN [2] Infinitesimal_HFinite_mult,
- THEN mem_infmal_iff [THEN iffD1]])
-apply (auto simp add: mult_commute
- intro: approx_trans approx_minus_iff [THEN iffD2])
-done
-
-text{*Differentiation rules for combinations of functions
- follow from clear, straightforard, algebraic
- manipulations*}
-text{*Constant function*}
-
-(* use simple constant nslimit theorem *)
-lemma NSDERIV_const [simp]: "(NSDERIV (%x. k) x :> 0)"
-by (simp add: NSDERIV_NSLIM_iff)
-
-text{*Sum of functions- proved easily*}
-
-lemma NSDERIV_add: "[| NSDERIV f x :> Da; NSDERIV g x :> Db |]
- ==> NSDERIV (%x. f x + g x) x :> Da + Db"
-apply (auto simp add: NSDERIV_NSLIM_iff NSLIM_def)
-apply (auto simp add: add_divide_distrib diff_divide_distrib dest!: spec)
-apply (drule_tac b = "hypreal_of_real Da" and d = "hypreal_of_real Db" in approx_add)
-apply (auto simp add: diff_def add_ac)
-done
-
-text{*Product of functions - Proof is trivial but tedious
- and long due to rearrangement of terms*}
-
-lemma lemma_nsderiv1: "((a::hypreal)*b) - (c*d) = (b*(a - c)) + (c*(b - d))"
-by (simp add: right_diff_distrib)
-
-lemma lemma_nsderiv2: "[| (x - y) / z = hypreal_of_real D + yb; z \<noteq> 0;
- z \<in> Infinitesimal; yb \<in> Infinitesimal |]
- ==> x - y \<approx> 0"
-apply (frule_tac c1 = z in hypreal_mult_right_cancel [THEN iffD2], assumption)
-apply (erule_tac V = "(x - y) / z = hypreal_of_real D + yb" in thin_rl)
-apply (auto intro!: Infinitesimal_HFinite_mult2 HFinite_add
- simp add: mult_assoc mem_infmal_iff [symmetric])
-apply (erule Infinitesimal_subset_HFinite [THEN subsetD])
-done
-
-lemma NSDERIV_mult: "[| NSDERIV f x :> Da; NSDERIV g x :> Db |]
- ==> NSDERIV (%x. f x * g x) x :> (Da * g(x)) + (Db * f(x))"
-apply (auto simp add: NSDERIV_NSLIM_iff NSLIM_def)
-apply (auto dest!: spec
- simp add: starfun_lambda_cancel lemma_nsderiv1)
-apply (simp (no_asm) add: add_divide_distrib diff_divide_distrib)
-apply (drule bex_Infinitesimal_iff2 [THEN iffD2])+
-apply (auto simp add: times_divide_eq_right [symmetric]
- simp del: times_divide_eq)
-apply (drule_tac D = Db in lemma_nsderiv2, assumption+)
-apply (drule_tac
- approx_minus_iff [THEN iffD2, THEN bex_Infinitesimal_iff2 [THEN iffD2]])
-apply (auto intro!: approx_add_mono1
- simp add: left_distrib right_distrib mult_commute add_assoc)
-apply (rule_tac b1 = "hypreal_of_real Db * hypreal_of_real (f x)"
- in add_commute [THEN subst])
-apply (auto intro!: Infinitesimal_add_approx_self2 [THEN approx_sym]
- Infinitesimal_add Infinitesimal_mult
- Infinitesimal_hypreal_of_real_mult
- Infinitesimal_hypreal_of_real_mult2
- simp add: add_assoc [symmetric])
-done
-
-text{*Multiplying by a constant*}
-lemma NSDERIV_cmult: "NSDERIV f x :> D
- ==> NSDERIV (%x. c * f x) x :> c*D"
-apply (simp only: times_divide_eq_right [symmetric] NSDERIV_NSLIM_iff
- minus_mult_right right_diff_distrib [symmetric])
-apply (erule NSLIM_const [THEN NSLIM_mult])
-done
-
-text{*Negation of function*}
-lemma NSDERIV_minus: "NSDERIV f x :> D ==> NSDERIV (%x. -(f x)) x :> -D"
-proof (simp add: NSDERIV_NSLIM_iff)
- assume "(\<lambda>h. (f (x + h) - f x) / h) -- 0 --NS> D"
- hence deriv: "(\<lambda>h. - ((f(x+h) - f x) / h)) -- 0 --NS> - D"
- by (rule NSLIM_minus)
- have "\<forall>h. - ((f (x + h) - f x) / h) = (- f (x + h) + f x) / h"
- by (simp add: minus_divide_left)
- with deriv
- show "(\<lambda>h. (- f (x + h) + f x) / h) -- 0 --NS> - D" by simp
-qed
-
-text{*Subtraction*}
-lemma NSDERIV_add_minus: "[| NSDERIV f x :> Da; NSDERIV g x :> Db |] ==> NSDERIV (%x. f x + -g x) x :> Da + -Db"
-by (blast dest: NSDERIV_add NSDERIV_minus)
-
-lemma NSDERIV_diff:
- "[| NSDERIV f x :> Da; NSDERIV g x :> Db |]
- ==> NSDERIV (%x. f x - g x) x :> Da-Db"
-apply (simp add: diff_minus)
-apply (blast intro: NSDERIV_add_minus)
-done
-
-text{* Similarly to the above, the chain rule admits an entirely
- straightforward derivation. Compare this with Harrison's
- HOL proof of the chain rule, which proved to be trickier and
- required an alternative characterisation of differentiability-
- the so-called Carathedory derivative. Our main problem is
- manipulation of terms.*}
-
-
-(* lemmas *)
-lemma NSDERIV_zero:
- "[| NSDERIV g x :> D;
- ( *f* g) (hypreal_of_real(x) + xa) = hypreal_of_real(g x);
- xa \<in> Infinitesimal;
- xa \<noteq> 0
- |] ==> D = 0"
-apply (simp add: nsderiv_def)
-apply (drule bspec, auto)
-done
-
-(* can be proved differently using NSLIM_isCont_iff *)
-lemma NSDERIV_approx:
- "[| NSDERIV f x :> D; h \<in> Infinitesimal; h \<noteq> 0 |]
- ==> ( *f* f) (hypreal_of_real(x) + h) - hypreal_of_real(f x) \<approx> 0"
-apply (simp add: nsderiv_def)
-apply (simp add: mem_infmal_iff [symmetric])
-apply (rule Infinitesimal_ratio)
-apply (rule_tac [3] approx_hypreal_of_real_HFinite, auto)
-done
-
-(*---------------------------------------------------------------
- from one version of differentiability
-
- f(x) - f(a)
- --------------- \<approx> Db
- x - a
- ---------------------------------------------------------------*)
-lemma NSDERIVD1: "[| NSDERIV f (g x) :> Da;
- ( *f* g) (hypreal_of_real(x) + xa) \<noteq> hypreal_of_real (g x);
- ( *f* g) (hypreal_of_real(x) + xa) \<approx> hypreal_of_real (g x)
- |] ==> (( *f* f) (( *f* g) (hypreal_of_real(x) + xa))
- - hypreal_of_real (f (g x)))
- / (( *f* g) (hypreal_of_real(x) + xa) - hypreal_of_real (g x))
- \<approx> hypreal_of_real(Da)"
-by (auto simp add: NSDERIV_NSLIM_iff2 NSLIM_def diff_minus [symmetric])
-
-(*--------------------------------------------------------------
- from other version of differentiability
-
- f(x + h) - f(x)
- ----------------- \<approx> Db
- h
- --------------------------------------------------------------*)
-lemma NSDERIVD2: "[| NSDERIV g x :> Db; xa \<in> Infinitesimal; xa \<noteq> 0 |]
- ==> (( *f* g) (hypreal_of_real(x) + xa) - hypreal_of_real(g x)) / xa
- \<approx> hypreal_of_real(Db)"
-by (auto simp add: NSDERIV_NSLIM_iff NSLIM_def mem_infmal_iff starfun_lambda_cancel)
-
-lemma lemma_chain: "(z::hypreal) \<noteq> 0 ==> x*y = (x*inverse(z))*(z*y)"
-by auto
-
-text{*This proof uses both definitions of differentiability.*}
-lemma NSDERIV_chain: "[| NSDERIV f (g x) :> Da; NSDERIV g x :> Db |]
- ==> NSDERIV (f o g) x :> Da * Db"
-apply (simp (no_asm_simp) add: NSDERIV_NSLIM_iff NSLIM_def
- mem_infmal_iff [symmetric])
-apply clarify
-apply (frule_tac f = g in NSDERIV_approx)
-apply (auto simp add: starfun_lambda_cancel2 starfun_o [symmetric])
-apply (case_tac "( *f* g) (hypreal_of_real (x) + xa) = hypreal_of_real (g x) ")
-apply (drule_tac g = g in NSDERIV_zero)
-apply (auto simp add: divide_inverse)
-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])
-apply (erule hypreal_not_eq_minus_iff [THEN iffD1])
-apply (rule approx_mult_hypreal_of_real)
-apply (simp_all add: divide_inverse [symmetric])
-apply (blast intro: NSDERIVD1 approx_minus_iff [THEN iffD2])
-apply (blast intro: NSDERIVD2)
-done
-
-text{*Differentiation of natural number powers*}
-lemma NSDERIV_Id [simp]: "NSDERIV (%x. x) x :> 1"
-by (simp add: NSDERIV_NSLIM_iff NSLIM_def divide_self del: divide_self_if)
-
-lemma NSDERIV_cmult_Id [simp]: "NSDERIV (op * c) x :> c"
-by (cut_tac c = c and x = x in NSDERIV_Id [THEN NSDERIV_cmult], simp)
-
-(*Can't get rid of x \<noteq> 0 because it isn't continuous at zero*)
-lemma NSDERIV_inverse:
- "x \<noteq> 0 ==> NSDERIV (%x. inverse(x)) x :> (- (inverse x ^ Suc (Suc 0)))"
-apply (simp add: nsderiv_def)
-apply (rule ballI, simp, clarify)
-apply (frule (1) Infinitesimal_add_not_zero)
-apply (simp add: add_commute)
-(*apply (auto simp add: starfun_inverse_inverse realpow_two
- simp del: minus_mult_left [symmetric] minus_mult_right [symmetric])*)
-apply (simp add: inverse_add inverse_mult_distrib [symmetric]
- inverse_minus_eq [symmetric] add_ac mult_ac diff_def
- del: inverse_mult_distrib inverse_minus_eq
- minus_mult_left [symmetric] minus_mult_right [symmetric])
-apply (simp (no_asm_simp) add: mult_assoc [symmetric] right_distrib
- del: minus_mult_left [symmetric] minus_mult_right [symmetric])
-apply (rule_tac y = "inverse (- hypreal_of_real x * hypreal_of_real x)" in approx_trans)
-apply (rule inverse_add_Infinitesimal_approx2)
-apply (auto dest!: hypreal_of_real_HFinite_diff_Infinitesimal
- simp add: inverse_minus_eq [symmetric] HFinite_minus_iff)
-apply (rule Infinitesimal_HFinite_mult2, auto)
-done
-
-subsubsection {* Equivalence of NS and Standard definitions *}
-
-lemma divideR_eq_divide [simp]: "x /# y = x / y"
-by (simp add: real_scaleR_def divide_inverse mult_commute)
-
-text{*Now equivalence between NSDERIV and DERIV*}
-lemma NSDERIV_DERIV_iff: "(NSDERIV f x :> D) = (DERIV f x :> D)"
-by (simp add: deriv_def NSDERIV_NSLIM_iff LIM_NSLIM_iff)
-
-(* let's do the standard proof though theorem *)
-(* LIM_mult2 follows from a NS proof *)
-
-lemma DERIV_cmult:
- fixes f :: "real \<Rightarrow> 'a::real_normed_algebra" shows
- "DERIV f x :> D ==> DERIV (%x. c * f x) x :> c*D"
-by (drule DERIV_mult' [OF DERIV_const], simp)
-
-(* standard version *)
-lemma DERIV_chain: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (f o g) x :> Da * Db"
-by (drule (1) DERIV_chain', simp add: o_def real_scaleR_def mult_commute)
-
-lemma DERIV_chain2: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (%x. f (g x)) x :> Da * Db"
-by (auto dest: DERIV_chain simp add: o_def)
-
-(*derivative of linear multiplication*)
-lemma DERIV_cmult_Id [simp]: "DERIV (op * c) x :> c"
-by (cut_tac c = c and x = x in DERIV_Id [THEN DERIV_cmult], simp)
-
-lemma DERIV_pow: "DERIV (%x. x ^ n) x :> real n * (x ^ (n - Suc 0))"
-apply (cut_tac DERIV_power [OF DERIV_Id])
-apply (simp add: real_scaleR_def real_of_nat_def)
-done
-
-(* NS version *)
-lemma NSDERIV_pow: "NSDERIV (%x. x ^ n) x :> real n * (x ^ (n - Suc 0))"
-by (simp add: NSDERIV_DERIV_iff DERIV_pow)
-
-text{*Power of -1*}
-
-lemma DERIV_inverse: "x \<noteq> 0 ==> DERIV (%x. inverse(x)) x :> (-(inverse x ^ Suc (Suc 0)))"
-by (drule DERIV_inverse' [OF DERIV_Id], simp)
-
-text{*Derivative of inverse*}
-lemma DERIV_inverse_fun: "[| DERIV f x :> d; f(x) \<noteq> 0 |]
- ==> DERIV (%x. inverse(f x)::real) x :> (- (d * inverse(f(x) ^ Suc (Suc 0))))"
-by (drule (1) DERIV_inverse', simp add: mult_ac)
-
-lemma NSDERIV_inverse_fun: "[| NSDERIV f x :> d; f(x) \<noteq> 0 |]
- ==> NSDERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ Suc (Suc 0))))"
-by (simp add: NSDERIV_DERIV_iff DERIV_inverse_fun del: realpow_Suc)
-
-text{*Derivative of quotient*}
-lemma DERIV_quotient: "[| DERIV f x :> d; DERIV g x :> e; g(x) \<noteq> 0 |]
- ==> DERIV (%y. f(y) / (g y) :: real) x :> (d*g(x) - (e*f(x))) / (g(x) ^ Suc (Suc 0))"
-by (drule (2) DERIV_divide, simp add: mult_commute)
-
-lemma NSDERIV_quotient: "[| NSDERIV f x :> d; DERIV g x :> e; g(x) \<noteq> 0 |]
- ==> NSDERIV (%y. f(y) / (g y)) x :> (d*g(x)
- - (e*f(x))) / (g(x) ^ Suc (Suc 0))"
-by (simp add: NSDERIV_DERIV_iff DERIV_quotient del: realpow_Suc)
-
-lemma CARAT_NSDERIV: "NSDERIV f x :> l ==>
- \<exists>g. (\<forall>z. f z - f x = g z * (z-x)) & isNSCont g x & g x = l"
-by (auto simp add: NSDERIV_DERIV_iff isNSCont_isCont_iff CARAT_DERIV
- real_scaleR_def mult_commute)
-
-lemma hypreal_eq_minus_iff3: "(x = y + z) = (x + -z = (y::hypreal))"
-by auto
-
-lemma CARAT_DERIVD:
- assumes all: "\<forall>z. f z - f x = g z * (z-x)"
- and nsc: "isNSCont g x"
- shows "NSDERIV f x :> g x"
-proof -
- from nsc
- have "\<forall>w. w \<noteq> hypreal_of_real x \<and> w \<approx> hypreal_of_real x \<longrightarrow>
- ( *f* g) w * (w - hypreal_of_real x) / (w - hypreal_of_real x) \<approx>
- hypreal_of_real (g x)"
- by (simp add: diff_minus isNSCont_def)
- thus ?thesis using all
- by (simp add: NSDERIV_iff2 starfun_if_eq cong: if_cong)
-qed
-
-subsubsection {* Differentiability predicate *}
-
-lemma differentiableD: "f differentiable x ==> \<exists>D. DERIV f x :> D"
-by (simp add: differentiable_def)
-
-lemma differentiableI: "DERIV f x :> D ==> f differentiable x"
-by (force simp add: differentiable_def)
-
-lemma NSdifferentiableD: "f NSdifferentiable x ==> \<exists>D. NSDERIV f x :> D"
-by (simp add: NSdifferentiable_def)
-
-lemma NSdifferentiableI: "NSDERIV f x :> D ==> f NSdifferentiable x"
-by (force simp add: NSdifferentiable_def)
-
-lemma differentiable_const: "(\<lambda>z. a) differentiable x"
- apply (unfold differentiable_def)
- apply (rule_tac x=0 in exI)
- apply simp
- done
-
-lemma differentiable_sum:
- assumes "f differentiable x"
- and "g differentiable x"
- shows "(\<lambda>x. f x + g x) differentiable x"
-proof -
- from prems have "\<exists>D. DERIV f x :> D" by (unfold differentiable_def)
- then obtain df where "DERIV f x :> df" ..
- moreover from prems have "\<exists>D. DERIV g x :> D" by (unfold differentiable_def)
- then obtain dg where "DERIV g x :> dg" ..
- ultimately have "DERIV (\<lambda>x. f x + g x) x :> df + dg" by (rule DERIV_add)
- hence "\<exists>D. DERIV (\<lambda>x. f x + g x) x :> D" by auto
- thus ?thesis by (fold differentiable_def)
-qed
-
-lemma differentiable_diff:
- assumes "f differentiable x"
- and "g differentiable x"
- shows "(\<lambda>x. f x - g x) differentiable x"
-proof -
- from prems have "f differentiable x" by simp
- moreover
- from prems have "\<exists>D. DERIV g x :> D" by (unfold differentiable_def)
- then obtain dg where "DERIV g x :> dg" ..
- then have "DERIV (\<lambda>x. - g x) x :> -dg" by (rule DERIV_minus)
- hence "\<exists>D. DERIV (\<lambda>x. - g x) x :> D" by auto
- hence "(\<lambda>x. - g x) differentiable x" by (fold differentiable_def)
- ultimately
- show ?thesis
- by (auto simp: real_diff_def dest: differentiable_sum)
-qed
-
-lemma differentiable_mult:
- assumes "f differentiable x"
- and "g differentiable x"
- shows "(\<lambda>x. f x * g x) differentiable x"
-proof -
- from prems have "\<exists>D. DERIV f x :> D" by (unfold differentiable_def)
- then obtain df where "DERIV f x :> df" ..
- moreover from prems have "\<exists>D. DERIV g x :> D" by (unfold differentiable_def)
- then obtain dg where "DERIV g x :> dg" ..
- ultimately have "DERIV (\<lambda>x. f x * g x) x :> df * g x + dg * f x" by (simp add: DERIV_mult)
- hence "\<exists>D. DERIV (\<lambda>x. f x * g x) x :> D" by auto
- thus ?thesis by (fold differentiable_def)
-qed
-
-subsection {*(NS) Increment*}
-lemma incrementI:
- "f NSdifferentiable x ==>
- increment f x h = ( *f* f) (hypreal_of_real(x) + h) -
- hypreal_of_real (f x)"
-by (simp add: increment_def)
-
-lemma incrementI2: "NSDERIV f x :> D ==>
- increment f x h = ( *f* f) (hypreal_of_real(x) + h) -
- hypreal_of_real (f x)"
-apply (erule NSdifferentiableI [THEN incrementI])
-done
-
-(* The Increment theorem -- Keisler p. 65 *)
-lemma increment_thm: "[| NSDERIV f x :> D; h \<in> Infinitesimal; h \<noteq> 0 |]
- ==> \<exists>e \<in> Infinitesimal. increment f x h = hypreal_of_real(D)*h + e*h"
-apply (frule_tac h = h in incrementI2, simp add: nsderiv_def)
-apply (drule bspec, auto)
-apply (drule bex_Infinitesimal_iff2 [THEN iffD2], clarify)
-apply (frule_tac b1 = "hypreal_of_real (D) + y"
- in hypreal_mult_right_cancel [THEN iffD2])
-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)
-apply assumption
-apply (simp add: times_divide_eq_right [symmetric])
-apply (auto simp add: left_distrib)
-done
-
-lemma increment_thm2:
- "[| NSDERIV f x :> D; h \<approx> 0; h \<noteq> 0 |]
- ==> \<exists>e \<in> Infinitesimal. increment f x h =
- hypreal_of_real(D)*h + e*h"
-by (blast dest!: mem_infmal_iff [THEN iffD2] intro!: increment_thm)
-
-
-lemma increment_approx_zero: "[| NSDERIV f x :> D; h \<approx> 0; h \<noteq> 0 |]
- ==> increment f x h \<approx> 0"
-apply (drule increment_thm2,
- auto intro!: Infinitesimal_HFinite_mult2 HFinite_add simp add: left_distrib [symmetric] mem_infmal_iff [symmetric])
-apply (erule Infinitesimal_subset_HFinite [THEN subsetD])
-done
-
-subsection {* Nested Intervals and Bisection *}
-
-text{*Lemmas about nested intervals and proof by bisection (cf.Harrison).
- All considerably tidied by lcp.*}
-
-lemma lemma_f_mono_add [rule_format (no_asm)]: "(\<forall>n. (f::nat=>real) n \<le> f (Suc n)) --> f m \<le> f(m + no)"
-apply (induct "no")
-apply (auto intro: order_trans)
-done
-
-lemma f_inc_g_dec_Beq_f: "[| \<forall>n. f(n) \<le> f(Suc n);
- \<forall>n. g(Suc n) \<le> g(n);
- \<forall>n. f(n) \<le> g(n) |]
- ==> Bseq (f :: nat \<Rightarrow> real)"
-apply (rule_tac k = "f 0" and K = "g 0" in BseqI2, rule allI)
-apply (induct_tac "n")
-apply (auto intro: order_trans)
-apply (rule_tac y = "g (Suc na)" in order_trans)
-apply (induct_tac [2] "na")
-apply (auto intro: order_trans)
-done
-
-lemma f_inc_g_dec_Beq_g: "[| \<forall>n. f(n) \<le> f(Suc n);
- \<forall>n. g(Suc n) \<le> g(n);
- \<forall>n. f(n) \<le> g(n) |]
- ==> Bseq (g :: nat \<Rightarrow> real)"
-apply (subst Bseq_minus_iff [symmetric])
-apply (rule_tac g = "%x. - (f x)" in f_inc_g_dec_Beq_f)
-apply auto
-done
-
-lemma f_inc_imp_le_lim:
- fixes f :: "nat \<Rightarrow> real"
- shows "\<lbrakk>\<forall>n. f n \<le> f (Suc n); convergent f\<rbrakk> \<Longrightarrow> f n \<le> lim f"
-apply (rule linorder_not_less [THEN iffD1])
-apply (auto simp add: convergent_LIMSEQ_iff LIMSEQ_iff monoseq_Suc)
-apply (drule real_less_sum_gt_zero)
-apply (drule_tac x = "f n + - lim f" in spec, safe)
-apply (drule_tac P = "%na. no\<le>na --> ?Q na" and x = "no + n" in spec, auto)
-apply (subgoal_tac "lim f \<le> f (no + n) ")
-apply (drule_tac no=no and m=n in lemma_f_mono_add)
-apply (auto simp add: add_commute)
-apply (induct_tac "no")
-apply simp
-apply (auto intro: order_trans simp add: diff_minus abs_if)
-done
-
-lemma lim_uminus: "convergent g ==> lim (%x. - g x) = - (lim g)"
-apply (rule LIMSEQ_minus [THEN limI])
-apply (simp add: convergent_LIMSEQ_iff)
-done
-
-lemma g_dec_imp_lim_le:
- fixes g :: "nat \<Rightarrow> real"
- shows "\<lbrakk>\<forall>n. g (Suc n) \<le> g(n); convergent g\<rbrakk> \<Longrightarrow> lim g \<le> g n"
-apply (subgoal_tac "- (g n) \<le> - (lim g) ")
-apply (cut_tac [2] f = "%x. - (g x)" in f_inc_imp_le_lim)
-apply (auto simp add: lim_uminus convergent_minus_iff [symmetric])
-done
-
-lemma lemma_nest: "[| \<forall>n. f(n) \<le> f(Suc n);
- \<forall>n. g(Suc n) \<le> g(n);
- \<forall>n. f(n) \<le> g(n) |]
- ==> \<exists>l m :: real. l \<le> m & ((\<forall>n. f(n) \<le> l) & f ----> l) &
- ((\<forall>n. m \<le> g(n)) & g ----> m)"
-apply (subgoal_tac "monoseq f & monoseq g")
-prefer 2 apply (force simp add: LIMSEQ_iff monoseq_Suc)
-apply (subgoal_tac "Bseq f & Bseq g")
-prefer 2 apply (blast intro: f_inc_g_dec_Beq_f f_inc_g_dec_Beq_g)
-apply (auto dest!: Bseq_monoseq_convergent simp add: convergent_LIMSEQ_iff)
-apply (rule_tac x = "lim f" in exI)
-apply (rule_tac x = "lim g" in exI)
-apply (auto intro: LIMSEQ_le)
-apply (auto simp add: f_inc_imp_le_lim g_dec_imp_lim_le convergent_LIMSEQ_iff)
-done
-
-lemma lemma_nest_unique: "[| \<forall>n. f(n) \<le> f(Suc n);
- \<forall>n. g(Suc n) \<le> g(n);
- \<forall>n. f(n) \<le> g(n);
- (%n. f(n) - g(n)) ----> 0 |]
- ==> \<exists>l::real. ((\<forall>n. f(n) \<le> l) & f ----> l) &
- ((\<forall>n. l \<le> g(n)) & g ----> l)"
-apply (drule lemma_nest, auto)
-apply (subgoal_tac "l = m")
-apply (drule_tac [2] X = f in LIMSEQ_diff)
-apply (auto intro: LIMSEQ_unique)
-done
-
-text{*The universal quantifiers below are required for the declaration
- of @{text Bolzano_nest_unique} below.*}
-
-lemma Bolzano_bisect_le:
- "a \<le> b ==> \<forall>n. fst (Bolzano_bisect P a b n) \<le> snd (Bolzano_bisect P a b n)"
-apply (rule allI)
-apply (induct_tac "n")
-apply (auto simp add: Let_def split_def)
-done
-
-lemma Bolzano_bisect_fst_le_Suc: "a \<le> b ==>
- \<forall>n. fst(Bolzano_bisect P a b n) \<le> fst (Bolzano_bisect P a b (Suc n))"
-apply (rule allI)
-apply (induct_tac "n")
-apply (auto simp add: Bolzano_bisect_le Let_def split_def)
-done
-
-lemma Bolzano_bisect_Suc_le_snd: "a \<le> b ==>
- \<forall>n. snd(Bolzano_bisect P a b (Suc n)) \<le> snd (Bolzano_bisect P a b n)"
-apply (rule allI)
-apply (induct_tac "n")
-apply (auto simp add: Bolzano_bisect_le Let_def split_def)
-done
-
-lemma eq_divide_2_times_iff: "((x::real) = y / (2 * z)) = (2 * x = y/z)"
-apply (auto)
-apply (drule_tac f = "%u. (1/2) *u" in arg_cong)
-apply (simp)
-done
-
-lemma Bolzano_bisect_diff:
- "a \<le> b ==>
- snd(Bolzano_bisect P a b n) - fst(Bolzano_bisect P a b n) =
- (b-a) / (2 ^ n)"
-apply (induct "n")
-apply (auto simp add: eq_divide_2_times_iff add_divide_distrib Let_def split_def)
-done
-
-lemmas Bolzano_nest_unique =
- lemma_nest_unique
- [OF Bolzano_bisect_fst_le_Suc Bolzano_bisect_Suc_le_snd Bolzano_bisect_le]
-
-
-lemma not_P_Bolzano_bisect:
- assumes P: "!!a b c. [| P(a,b); P(b,c); a \<le> b; b \<le> c|] ==> P(a,c)"
- and notP: "~ P(a,b)"
- and le: "a \<le> b"
- shows "~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))"
-proof (induct n)
- case 0 thus ?case by simp
- next
- case (Suc n)
- thus ?case
- by (auto simp del: surjective_pairing [symmetric]
- simp add: Let_def split_def Bolzano_bisect_le [OF le]
- P [of "fst (Bolzano_bisect P a b n)" _ "snd (Bolzano_bisect P a b n)"])
-qed
-
-(*Now we re-package P_prem as a formula*)
-lemma not_P_Bolzano_bisect':
- "[| \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c --> P(a,c);
- ~ P(a,b); a \<le> b |] ==>
- \<forall>n. ~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))"
-by (blast elim!: not_P_Bolzano_bisect [THEN [2] rev_notE])
-
-
-
-lemma lemma_BOLZANO:
- "[| \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c --> P(a,c);
- \<forall>x. \<exists>d::real. 0 < d &
- (\<forall>a b. a \<le> x & x \<le> b & (b-a) < d --> P(a,b));
- a \<le> b |]
- ==> P(a,b)"
-apply (rule Bolzano_nest_unique [where P1=P, THEN exE], assumption+)
-apply (rule LIMSEQ_minus_cancel)
-apply (simp (no_asm_simp) add: Bolzano_bisect_diff LIMSEQ_divide_realpow_zero)
-apply (rule ccontr)
-apply (drule not_P_Bolzano_bisect', assumption+)
-apply (rename_tac "l")
-apply (drule_tac x = l in spec, clarify)
-apply (simp add: LIMSEQ_def)
-apply (drule_tac P = "%r. 0<r --> ?Q r" and x = "d/2" in spec)
-apply (drule_tac P = "%r. 0<r --> ?Q r" and x = "d/2" in spec)
-apply (drule real_less_half_sum, auto)
-apply (drule_tac x = "fst (Bolzano_bisect P a b (no + noa))" in spec)
-apply (drule_tac x = "snd (Bolzano_bisect P a b (no + noa))" in spec)
-apply safe
-apply (simp_all (no_asm_simp))
-apply (rule_tac y = "abs (fst (Bolzano_bisect P a b (no + noa)) - l) + abs (snd (Bolzano_bisect P a b (no + noa)) - l)" in order_le_less_trans)
-apply (simp (no_asm_simp) add: abs_if)
-apply (rule real_sum_of_halves [THEN subst])
-apply (rule add_strict_mono)
-apply (simp_all add: diff_minus [symmetric])
-done
-
-
-lemma lemma_BOLZANO2: "((\<forall>a b c. (a \<le> b & b \<le> c & P(a,b) & P(b,c)) --> P(a,c)) &
- (\<forall>x. \<exists>d::real. 0 < d &
- (\<forall>a b. a \<le> x & x \<le> b & (b-a) < d --> P(a,b))))
- --> (\<forall>a b. a \<le> b --> P(a,b))"
-apply clarify
-apply (blast intro: lemma_BOLZANO)
-done
-
-
-subsection {* Intermediate Value Theorem *}
-
-text {*Prove Contrapositive by Bisection*}
-
-lemma IVT: "[| f(a::real) \<le> (y::real); y \<le> f(b);
- a \<le> b;
- (\<forall>x. a \<le> x & x \<le> b --> isCont f x) |]
- ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y"
-apply (rule contrapos_pp, assumption)
-apply (cut_tac P = "% (u,v) . a \<le> u & u \<le> v & v \<le> b --> ~ (f (u) \<le> y & y \<le> f (v))" in lemma_BOLZANO2)
-apply safe
-apply simp_all
-apply (simp add: isCont_iff LIM_def)
-apply (rule ccontr)
-apply (subgoal_tac "a \<le> x & x \<le> b")
- prefer 2
- apply simp
- apply (drule_tac P = "%d. 0<d --> ?P d" and x = 1 in spec, arith)
-apply (drule_tac x = x in spec)+
-apply simp
-apply (drule_tac P = "%r. ?P r --> (\<exists>s>0. ?Q r s) " and x = "\<bar>y - f x\<bar>" in spec)
-apply safe
-apply simp
-apply (drule_tac x = s in spec, clarify)
-apply (cut_tac x = "f x" and y = y in linorder_less_linear, safe)
-apply (drule_tac x = "ba-x" in spec)
-apply (simp_all add: abs_if)
-apply (drule_tac x = "aa-x" in spec)
-apply (case_tac "x \<le> aa", simp_all)
-done
-
-lemma IVT2: "[| f(b::real) \<le> (y::real); y \<le> f(a);
- a \<le> b;
- (\<forall>x. a \<le> x & x \<le> b --> isCont f x)
- |] ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y"
-apply (subgoal_tac "- f a \<le> -y & -y \<le> - f b", clarify)
-apply (drule IVT [where f = "%x. - f x"], assumption)
-apply (auto intro: isCont_minus)
-done
-
-(*HOL style here: object-level formulations*)
-lemma IVT_objl: "(f(a::real) \<le> (y::real) & y \<le> f(b) & a \<le> b &
- (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
- --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
-apply (blast intro: IVT)
-done
-
-lemma IVT2_objl: "(f(b::real) \<le> (y::real) & y \<le> f(a) & a \<le> b &
- (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
- --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
-apply (blast intro: IVT2)
-done
-
-text{*By bisection, function continuous on closed interval is bounded above*}
-
-lemma isCont_bounded:
- "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
- ==> \<exists>M::real. \<forall>x::real. a \<le> x & x \<le> b --> f(x) \<le> M"
-apply (cut_tac P = "% (u,v) . a \<le> u & u \<le> v & v \<le> b --> (\<exists>M. \<forall>x. u \<le> x & x \<le> v --> f x \<le> M)" in lemma_BOLZANO2)
-apply safe
-apply simp_all
-apply (rename_tac x xa ya M Ma)
-apply (cut_tac x = M and y = Ma in linorder_linear, safe)
-apply (rule_tac x = Ma in exI, clarify)
-apply (cut_tac x = xb and y = xa in linorder_linear, force)
-apply (rule_tac x = M in exI, clarify)
-apply (cut_tac x = xb and y = xa in linorder_linear, force)
-apply (case_tac "a \<le> x & x \<le> b")
-apply (rule_tac [2] x = 1 in exI)
-prefer 2 apply force
-apply (simp add: LIM_def isCont_iff)
-apply (drule_tac x = x in spec, auto)
-apply (erule_tac V = "\<forall>M. \<exists>x. a \<le> x & x \<le> b & ~ f x \<le> M" in thin_rl)
-apply (drule_tac x = 1 in spec, auto)
-apply (rule_tac x = s in exI, clarify)
-apply (rule_tac x = "\<bar>f x\<bar> + 1" in exI, clarify)
-apply (drule_tac x = "xa-x" in spec)
-apply (auto simp add: abs_ge_self)
-done
-
-text{*Refine the above to existence of least upper bound*}
-
-lemma lemma_reals_complete: "((\<exists>x. x \<in> S) & (\<exists>y. isUb UNIV S (y::real))) -->
- (\<exists>t. isLub UNIV S t)"
-by (blast intro: reals_complete)
-
-lemma isCont_has_Ub: "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
- ==> \<exists>M::real. (\<forall>x::real. a \<le> x & x \<le> b --> f(x) \<le> M) &
- (\<forall>N. N < M --> (\<exists>x. a \<le> x & x \<le> b & N < f(x)))"
-apply (cut_tac S = "Collect (%y. \<exists>x. a \<le> x & x \<le> b & y = f x)"
- in lemma_reals_complete)
-apply auto
-apply (drule isCont_bounded, assumption)
-apply (auto simp add: isUb_def leastP_def isLub_def setge_def setle_def)
-apply (rule exI, auto)
-apply (auto dest!: spec simp add: linorder_not_less)
-done
-
-text{*Now show that it attains its upper bound*}
-
-lemma isCont_eq_Ub:
- assumes le: "a \<le> b"
- and con: "\<forall>x::real. a \<le> x & x \<le> b --> isCont f x"
- shows "\<exists>M::real. (\<forall>x. a \<le> x & x \<le> b --> f(x) \<le> M) &
- (\<exists>x. a \<le> x & x \<le> b & f(x) = M)"
-proof -
- from isCont_has_Ub [OF le con]
- obtain M where M1: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M"
- and M2: "!!N. N<M ==> \<exists>x. a \<le> x \<and> x \<le> b \<and> N < f x" by blast
- show ?thesis
- proof (intro exI, intro conjI)
- show " \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M" by (rule M1)
- show "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M"
- proof (rule ccontr)
- assume "\<not> (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M)"
- with M1 have M3: "\<forall>x. a \<le> x & x \<le> b --> f x < M"
- by (fastsimp simp add: linorder_not_le [symmetric])
- hence "\<forall>x. a \<le> x & x \<le> b --> isCont (%x. inverse (M - f x)) x"
- by (auto simp add: isCont_inverse isCont_diff con)
- from isCont_bounded [OF le this]
- obtain k where k: "!!x. a \<le> x & x \<le> b --> inverse (M - f x) \<le> k" by auto
- have Minv: "!!x. a \<le> x & x \<le> b --> 0 < inverse (M - f (x))"
- by (simp add: M3 compare_rls)
- have "!!x. a \<le> x & x \<le> b --> inverse (M - f x) < k+1" using k
- by (auto intro: order_le_less_trans [of _ k])
- with Minv
- have "!!x. a \<le> x & x \<le> b --> inverse(k+1) < inverse(inverse(M - f x))"
- by (intro strip less_imp_inverse_less, simp_all)
- hence invlt: "!!x. a \<le> x & x \<le> b --> inverse(k+1) < M - f x"
- by simp
- have "M - inverse (k+1) < M" using k [of a] Minv [of a] le
- by (simp, arith)
- from M2 [OF this]
- obtain x where ax: "a \<le> x & x \<le> b & M - inverse(k+1) < f x" ..
- thus False using invlt [of x] by force
- qed
- qed
-qed
-
-
-text{*Same theorem for lower bound*}
-
-lemma isCont_eq_Lb: "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
- ==> \<exists>M::real. (\<forall>x::real. a \<le> x & x \<le> b --> M \<le> f(x)) &
- (\<exists>x. a \<le> x & x \<le> b & f(x) = M)"
-apply (subgoal_tac "\<forall>x. a \<le> x & x \<le> b --> isCont (%x. - (f x)) x")
-prefer 2 apply (blast intro: isCont_minus)
-apply (drule_tac f = "(%x. - (f x))" in isCont_eq_Ub)
-apply safe
-apply auto
-done
-
-
-text{*Another version.*}
-
-lemma isCont_Lb_Ub: "[|a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
- ==> \<exists>L M::real. (\<forall>x::real. a \<le> x & x \<le> b --> L \<le> f(x) & f(x) \<le> M) &
- (\<forall>y. L \<le> y & y \<le> M --> (\<exists>x. a \<le> x & x \<le> b & (f(x) = y)))"
-apply (frule isCont_eq_Lb)
-apply (frule_tac [2] isCont_eq_Ub)
-apply (assumption+, safe)
-apply (rule_tac x = "f x" in exI)
-apply (rule_tac x = "f xa" in exI, simp, safe)
-apply (cut_tac x = x and y = xa in linorder_linear, safe)
-apply (cut_tac f = f and a = x and b = xa and y = y in IVT_objl)
-apply (cut_tac [2] f = f and a = xa and b = x and y = y in IVT2_objl, safe)
-apply (rule_tac [2] x = xb in exI)
-apply (rule_tac [4] x = xb in exI, simp_all)
-done
-
-
-text{*If @{term "0 < f'(x)"} then @{term x} is Locally Strictly Increasing At The Right*}
-
-lemma DERIV_left_inc:
- fixes f :: "real => real"
- assumes der: "DERIV f x :> l"
- and l: "0 < l"
- shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x + h)"
-proof -
- from l der [THEN DERIV_D, THEN LIM_D [where r = "l"]]
- have "\<exists>s > 0. (\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < l)"
- by (simp add: diff_minus)
- then obtain s
- where s: "0 < s"
- and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < l"
- by auto
- thus ?thesis
- proof (intro exI conjI strip)
- show "0<s" .
- fix h::real
- assume "0 < h" "h < s"
- with all [of h] show "f x < f (x+h)"
- proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric]
- split add: split_if_asm)
- assume "~ (f (x+h) - f x) / h < l" and h: "0 < h"
- with l
- have "0 < (f (x+h) - f x) / h" by arith
- thus "f x < f (x+h)"
- by (simp add: pos_less_divide_eq h)
- qed
- qed
-qed
-
-lemma DERIV_left_dec:
- fixes f :: "real => real"
- assumes der: "DERIV f x :> l"
- and l: "l < 0"
- shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x-h)"
-proof -
- from l der [THEN DERIV_D, THEN LIM_D [where r = "-l"]]
- have "\<exists>s > 0. (\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < -l)"
- by (simp add: diff_minus)
- then obtain s
- where s: "0 < s"
- and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < -l"
- by auto
- thus ?thesis
- proof (intro exI conjI strip)
- show "0<s" .
- fix h::real
- assume "0 < h" "h < s"
- with all [of "-h"] show "f x < f (x-h)"
- proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric]
- split add: split_if_asm)
- assume " - ((f (x-h) - f x) / h) < l" and h: "0 < h"
- with l
- have "0 < (f (x-h) - f x) / h" by arith
- thus "f x < f (x-h)"
- by (simp add: pos_less_divide_eq h)
- qed
- qed
-qed
-
-lemma DERIV_local_max:
- fixes f :: "real => real"
- assumes der: "DERIV f x :> l"
- and d: "0 < d"
- and le: "\<forall>y. \<bar>x-y\<bar> < d --> f(y) \<le> f(x)"
- shows "l = 0"
-proof (cases rule: linorder_cases [of l 0])
- case equal show ?thesis .
-next
- case less
- from DERIV_left_dec [OF der less]
- obtain d' where d': "0 < d'"
- and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x-h)" by blast
- from real_lbound_gt_zero [OF d d']
- obtain e where "0 < e \<and> e < d \<and> e < d'" ..
- with lt le [THEN spec [where x="x-e"]]
- show ?thesis by (auto simp add: abs_if)
-next
- case greater
- from DERIV_left_inc [OF der greater]
- obtain d' where d': "0 < d'"
- and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x + h)" by blast
- from real_lbound_gt_zero [OF d d']
- obtain e where "0 < e \<and> e < d \<and> e < d'" ..
- with lt le [THEN spec [where x="x+e"]]
- show ?thesis by (auto simp add: abs_if)
-qed
-
-
-text{*Similar theorem for a local minimum*}
-lemma DERIV_local_min:
- fixes f :: "real => real"
- shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) \<le> f(y) |] ==> l = 0"
-by (drule DERIV_minus [THEN DERIV_local_max], auto)
-
-
-text{*In particular, if a function is locally flat*}
-lemma DERIV_local_const:
- fixes f :: "real => real"
- shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) = f(y) |] ==> l = 0"
-by (auto dest!: DERIV_local_max)
-
-text{*Lemma about introducing open ball in open interval*}
-lemma lemma_interval_lt:
- "[| a < x; x < b |]
- ==> \<exists>d::real. 0 < d & (\<forall>y. \<bar>x-y\<bar> < d --> a < y & y < b)"
-apply (simp add: abs_interval_iff)
-apply (insert linorder_linear [of "x-a" "b-x"], safe)
-apply (rule_tac x = "x-a" in exI)
-apply (rule_tac [2] x = "b-x" in exI, auto)
-done
-
-lemma lemma_interval: "[| a < x; x < b |] ==>
- \<exists>d::real. 0 < d & (\<forall>y. \<bar>x-y\<bar> < d --> a \<le> y & y \<le> b)"
-apply (drule lemma_interval_lt, auto)
-apply (auto intro!: exI)
-done
-
-text{*Rolle's Theorem.
- If @{term f} is defined and continuous on the closed interval
- @{text "[a,b]"} and differentiable on the open interval @{text "(a,b)"},
- and @{term "f(a) = f(b)"},
- then there exists @{text "x0 \<in> (a,b)"} such that @{term "f'(x0) = 0"}*}
-theorem Rolle:
- assumes lt: "a < b"
- and eq: "f(a) = f(b)"
- and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
- and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"
- shows "\<exists>z. a < z & z < b & DERIV f z :> 0"
-proof -
- have le: "a \<le> b" using lt by simp
- from isCont_eq_Ub [OF le con]
- obtain x where x_max: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f z \<le> f x"
- and alex: "a \<le> x" and xleb: "x \<le> b"
- by blast
- from isCont_eq_Lb [OF le con]
- obtain x' where x'_min: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f x' \<le> f z"
- and alex': "a \<le> x'" and x'leb: "x' \<le> b"
- by blast
- show ?thesis
- proof cases
- assume axb: "a < x & x < b"
- --{*@{term f} attains its maximum within the interval*}
- hence ax: "a<x" and xb: "x<b" by auto
- from lemma_interval [OF ax xb]
- obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
- by blast
- hence bound': "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> f y \<le> f x" using x_max
- by blast
- from differentiableD [OF dif [OF axb]]
- obtain l where der: "DERIV f x :> l" ..
- have "l=0" by (rule DERIV_local_max [OF der d bound'])
- --{*the derivative at a local maximum is zero*}
- thus ?thesis using ax xb der by auto
- next
- assume notaxb: "~ (a < x & x < b)"
- hence xeqab: "x=a | x=b" using alex xleb by arith
- hence fb_eq_fx: "f b = f x" by (auto simp add: eq)
- show ?thesis
- proof cases
- assume ax'b: "a < x' & x' < b"
- --{*@{term f} attains its minimum within the interval*}
- hence ax': "a<x'" and x'b: "x'<b" by auto
- from lemma_interval [OF ax' x'b]
- obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
- by blast
- hence bound': "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> f x' \<le> f y" using x'_min
- by blast
- from differentiableD [OF dif [OF ax'b]]
- obtain l where der: "DERIV f x' :> l" ..
- have "l=0" by (rule DERIV_local_min [OF der d bound'])
- --{*the derivative at a local minimum is zero*}
- thus ?thesis using ax' x'b der by auto
- next
- assume notax'b: "~ (a < x' & x' < b)"
- --{*@{term f} is constant througout the interval*}
- hence x'eqab: "x'=a | x'=b" using alex' x'leb by arith
- hence fb_eq_fx': "f b = f x'" by (auto simp add: eq)
- from dense [OF lt]
- obtain r where ar: "a < r" and rb: "r < b" by blast
- from lemma_interval [OF ar rb]
- obtain d where d: "0<d" and bound: "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
- by blast
- have eq_fb: "\<forall>z. a \<le> z --> z \<le> b --> f z = f b"
- proof (clarify)
- fix z::real
- assume az: "a \<le> z" and zb: "z \<le> b"
- show "f z = f b"
- proof (rule order_antisym)
- show "f z \<le> f b" by (simp add: fb_eq_fx x_max az zb)
- show "f b \<le> f z" by (simp add: fb_eq_fx' x'_min az zb)
- qed
- qed
- have bound': "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> f r = f y"
- proof (intro strip)
- fix y::real
- assume lt: "\<bar>r-y\<bar> < d"
- hence "f y = f b" by (simp add: eq_fb bound)
- thus "f r = f y" by (simp add: eq_fb ar rb order_less_imp_le)
- qed
- from differentiableD [OF dif [OF conjI [OF ar rb]]]
- obtain l where der: "DERIV f r :> l" ..
- have "l=0" by (rule DERIV_local_const [OF der d bound'])
- --{*the derivative of a constant function is zero*}
- thus ?thesis using ar rb der by auto
- qed
- qed
-qed
-
-
-subsection{*Mean Value Theorem*}
-
-lemma lemma_MVT:
- "f a - (f b - f a)/(b-a) * a = f b - (f b - f a)/(b-a) * (b::real)"
-proof cases
- assume "a=b" thus ?thesis by simp
-next
- assume "a\<noteq>b"
- hence ba: "b-a \<noteq> 0" by arith
- show ?thesis
- by (rule real_mult_left_cancel [OF ba, THEN iffD1],
- simp add: right_diff_distrib,
- simp add: left_diff_distrib)
-qed
-
-theorem MVT:
- assumes lt: "a < b"
- and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
- and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"
- shows "\<exists>l z. a < z & z < b & DERIV f z :> l &
- (f(b) - f(a) = (b-a) * l)"
-proof -
- let ?F = "%x. f x - ((f b - f a) / (b-a)) * x"
- have contF: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?F x" using con
- by (fast intro: isCont_diff isCont_const isCont_mult isCont_Id)
- have difF: "\<forall>x. a < x \<and> x < b \<longrightarrow> ?F differentiable x"
- proof (clarify)
- fix x::real
- assume ax: "a < x" and xb: "x < b"
- from differentiableD [OF dif [OF conjI [OF ax xb]]]
- obtain l where der: "DERIV f x :> l" ..
- show "?F differentiable x"
- by (rule differentiableI [where D = "l - (f b - f a)/(b-a)"],
- blast intro: DERIV_diff DERIV_cmult_Id der)
- qed
- from Rolle [where f = ?F, OF lt lemma_MVT contF difF]
- obtain z where az: "a < z" and zb: "z < b" and der: "DERIV ?F z :> 0"
- by blast
- have "DERIV (%x. ((f b - f a)/(b-a)) * x) z :> (f b - f a)/(b-a)"
- by (rule DERIV_cmult_Id)
- hence derF: "DERIV (\<lambda>x. ?F x + (f b - f a) / (b - a) * x) z
- :> 0 + (f b - f a) / (b - a)"
- by (rule DERIV_add [OF der])
- show ?thesis
- proof (intro exI conjI)
- show "a < z" .
- show "z < b" .
- show "f b - f a = (b - a) * ((f b - f a)/(b-a))" by (simp)
- show "DERIV f z :> ((f b - f a)/(b-a))" using derF by simp
- qed
-qed
-
-
-text{*A function is constant if its derivative is 0 over an interval.*}
-
-lemma DERIV_isconst_end:
- fixes f :: "real => real"
- shows "[| a < b;
- \<forall>x. a \<le> x & x \<le> b --> isCont f x;
- \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
- ==> f b = f a"
-apply (drule MVT, assumption)
-apply (blast intro: differentiableI)
-apply (auto dest!: DERIV_unique simp add: diff_eq_eq)
-done
-
-lemma DERIV_isconst1:
- fixes f :: "real => real"
- shows "[| a < b;
- \<forall>x. a \<le> x & x \<le> b --> isCont f x;
- \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
- ==> \<forall>x. a \<le> x & x \<le> b --> f x = f a"
-apply safe
-apply (drule_tac x = a in order_le_imp_less_or_eq, safe)
-apply (drule_tac b = x in DERIV_isconst_end, auto)
-done
-
-lemma DERIV_isconst2:
- fixes f :: "real => real"
- shows "[| a < b;
- \<forall>x. a \<le> x & x \<le> b --> isCont f x;
- \<forall>x. a < x & x < b --> DERIV f x :> 0;
- a \<le> x; x \<le> b |]
- ==> f x = f a"
-apply (blast dest: DERIV_isconst1)
-done
-
-lemma DERIV_isconst_all:
- fixes f :: "real => real"
- shows "\<forall>x. DERIV f x :> 0 ==> f(x) = f(y)"
-apply (rule linorder_cases [of x y])
-apply (blast intro: sym DERIV_isCont DERIV_isconst_end)+
-done
-
-lemma DERIV_const_ratio_const:
- "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a)) = (b-a) * k"
-apply (rule linorder_cases [of a b], auto)
-apply (drule_tac [!] f = f in MVT)
-apply (auto dest: DERIV_isCont DERIV_unique simp add: differentiable_def)
-apply (auto dest: DERIV_unique simp add: left_distrib diff_minus)
-done
-
-lemma DERIV_const_ratio_const2:
- "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a))/(b-a) = k"
-apply (rule_tac c1 = "b-a" in real_mult_right_cancel [THEN iffD1])
-apply (auto dest!: DERIV_const_ratio_const simp add: mult_assoc)
-done
-
-lemma real_average_minus_first [simp]: "((a + b) /2 - a) = (b-a)/(2::real)"
-by (simp)
-
-lemma real_average_minus_second [simp]: "((b + a)/2 - a) = (b-a)/(2::real)"
-by (simp)
-
-text{*Gallileo's "trick": average velocity = av. of end velocities*}
-
-lemma DERIV_const_average:
- fixes v :: "real => real"
- assumes neq: "a \<noteq> (b::real)"
- and der: "\<forall>x. DERIV v x :> k"
- shows "v ((a + b)/2) = (v a + v b)/2"
-proof (cases rule: linorder_cases [of a b])
- case equal with neq show ?thesis by simp
-next
- case less
- have "(v b - v a) / (b - a) = k"
- by (rule DERIV_const_ratio_const2 [OF neq der])
- hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
- moreover have "(v ((a + b) / 2) - v a) / ((a + b) / 2 - a) = k"
- by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
- ultimately show ?thesis using neq by force
-next
- case greater
- have "(v b - v a) / (b - a) = k"
- by (rule DERIV_const_ratio_const2 [OF neq der])
- hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
- moreover have " (v ((b + a) / 2) - v a) / ((b + a) / 2 - a) = k"
- by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
- ultimately show ?thesis using neq by (force simp add: add_commute)
-qed
-
-
-text{*Dull lemma: an continuous injection on an interval must have a
-strict maximum at an end point, not in the middle.*}
-
-lemma lemma_isCont_inj:
- fixes f :: "real \<Rightarrow> real"
- assumes d: "0 < d"
- and inj [rule_format]: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
- and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
- shows "\<exists>z. \<bar>z-x\<bar> \<le> d & f x < f z"
-proof (rule ccontr)
- assume "~ (\<exists>z. \<bar>z-x\<bar> \<le> d & f x < f z)"
- hence all [rule_format]: "\<forall>z. \<bar>z - x\<bar> \<le> d --> f z \<le> f x" by auto
- show False
- proof (cases rule: linorder_le_cases [of "f(x-d)" "f(x+d)"])
- case le
- from d cont all [of "x+d"]
- have flef: "f(x+d) \<le> f x"
- and xlex: "x - d \<le> x"
- and cont': "\<forall>z. x - d \<le> z \<and> z \<le> x \<longrightarrow> isCont f z"
- by (auto simp add: abs_if)
- from IVT [OF le flef xlex cont']
- obtain x' where "x-d \<le> x'" "x' \<le> x" "f x' = f(x+d)" by blast
- moreover
- hence "g(f x') = g (f(x+d))" by simp
- ultimately show False using d inj [of x'] inj [of "x+d"]
- by (simp add: abs_le_interval_iff)
- next
- case ge
- from d cont all [of "x-d"]
- have flef: "f(x-d) \<le> f x"
- and xlex: "x \<le> x+d"
- and cont': "\<forall>z. x \<le> z \<and> z \<le> x+d \<longrightarrow> isCont f z"
- by (auto simp add: abs_if)
- from IVT2 [OF ge flef xlex cont']
- obtain x' where "x \<le> x'" "x' \<le> x+d" "f x' = f(x-d)" by blast
- moreover
- hence "g(f x') = g (f(x-d))" by simp
- ultimately show False using d inj [of x'] inj [of "x-d"]
- by (simp add: abs_le_interval_iff)
- qed
-qed
-
-
-text{*Similar version for lower bound.*}
-
-lemma lemma_isCont_inj2:
- fixes f g :: "real \<Rightarrow> real"
- shows "[|0 < d; \<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z;
- \<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z |]
- ==> \<exists>z. \<bar>z-x\<bar> \<le> d & f z < f x"
-apply (insert lemma_isCont_inj
- [where f = "%x. - f x" and g = "%y. g(-y)" and x = x and d = d])
-apply (simp add: isCont_minus linorder_not_le)
-done
-
-text{*Show there's an interval surrounding @{term "f(x)"} in
-@{text "f[[x - d, x + d]]"} .*}
-
-lemma isCont_inj_range:
- fixes f :: "real \<Rightarrow> real"
- assumes d: "0 < d"
- and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
- and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
- shows "\<exists>e>0. \<forall>y. \<bar>y - f x\<bar> \<le> e --> (\<exists>z. \<bar>z-x\<bar> \<le> d & f z = y)"
-proof -
- have "x-d \<le> x+d" "\<forall>z. x-d \<le> z \<and> z \<le> x+d \<longrightarrow> isCont f z" using cont d
- by (auto simp add: abs_le_interval_iff)
- from isCont_Lb_Ub [OF this]
- obtain L M
- where all1 [rule_format]: "\<forall>z. x-d \<le> z \<and> z \<le> x+d \<longrightarrow> L \<le> f z \<and> f z \<le> M"
- and all2 [rule_format]:
- "\<forall>y. L \<le> y \<and> y \<le> M \<longrightarrow> (\<exists>z. x-d \<le> z \<and> z \<le> x+d \<and> f z = y)"
- by auto
- with d have "L \<le> f x & f x \<le> M" by simp
- moreover have "L \<noteq> f x"
- proof -
- from lemma_isCont_inj2 [OF d inj cont]
- obtain u where "\<bar>u - x\<bar> \<le> d" "f u < f x" by auto
- thus ?thesis using all1 [of u] by arith
- qed
- moreover have "f x \<noteq> M"
- proof -
- from lemma_isCont_inj [OF d inj cont]
- obtain u where "\<bar>u - x\<bar> \<le> d" "f x < f u" by auto
- thus ?thesis using all1 [of u] by arith
- qed
- ultimately have "L < f x & f x < M" by arith
- hence "0 < f x - L" "0 < M - f x" by arith+
- from real_lbound_gt_zero [OF this]
- obtain e where e: "0 < e" "e < f x - L" "e < M - f x" by auto
- thus ?thesis
- proof (intro exI conjI)
- show "0<e" .
- show "\<forall>y. \<bar>y - f x\<bar> \<le> e \<longrightarrow> (\<exists>z. \<bar>z - x\<bar> \<le> d \<and> f z = y)"
- proof (intro strip)
- fix y::real
- assume "\<bar>y - f x\<bar> \<le> e"
- with e have "L \<le> y \<and> y \<le> M" by arith
- from all2 [OF this]
- obtain z where "x - d \<le> z" "z \<le> x + d" "f z = y" by blast
- thus "\<exists>z. \<bar>z - x\<bar> \<le> d \<and> f z = y"
- by (force simp add: abs_le_interval_iff)
- qed
- qed
-qed
-
-
-text{*Continuity of inverse function*}
-
-lemma isCont_inverse_function:
- fixes f g :: "real \<Rightarrow> real"
- assumes d: "0 < d"
- and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
- and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
- shows "isCont g (f x)"
-proof (simp add: isCont_iff LIM_eq)
- show "\<forall>r. 0 < r \<longrightarrow>
- (\<exists>s>0. \<forall>z. z\<noteq>0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>g(f x + z) - g(f x)\<bar> < r)"
- proof (intro strip)
- fix r::real
- assume r: "0<r"
- from real_lbound_gt_zero [OF r d]
- obtain e where e: "0 < e" and e_lt: "e < r \<and> e < d" by blast
- with inj cont
- have e_simps: "\<forall>z. \<bar>z-x\<bar> \<le> e --> g (f z) = z"
- "\<forall>z. \<bar>z-x\<bar> \<le> e --> isCont f z" by auto
- from isCont_inj_range [OF e this]
- obtain e' where e': "0 < e'"
- and all: "\<forall>y. \<bar>y - f x\<bar> \<le> e' \<longrightarrow> (\<exists>z. \<bar>z - x\<bar> \<le> e \<and> f z = y)"
- by blast
- show "\<exists>s>0. \<forall>z. z\<noteq>0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>g(f x + z) - g(f x)\<bar> < r"
- proof (intro exI conjI)
- show "0<e'" .
- show "\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < e' \<longrightarrow> \<bar>g (f x + z) - g (f x)\<bar> < r"
- proof (intro strip)
- fix z::real
- assume z: "z \<noteq> 0 \<and> \<bar>z\<bar> < e'"
- with e e_lt e_simps all [rule_format, of "f x + z"]
- show "\<bar>g (f x + z) - g (f x)\<bar> < r" by force
- qed
- qed
- qed
-qed
-
-theorem GMVT:
- assumes alb: "a < b"
- and fc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
- and fd: "\<forall>x. a < x \<and> x < b \<longrightarrow> f differentiable x"
- and gc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont g x"
- and gd: "\<forall>x. a < x \<and> x < b \<longrightarrow> g differentiable x"
- shows "\<exists>g'c f'c c. DERIV g c :> g'c \<and> DERIV f c :> f'c \<and> a < c \<and> c < b \<and> ((f b - f a) * g'c) = ((g b - g a) * f'c)"
-proof -
- let ?h = "\<lambda>x. (f b - f a)*(g x) - (g b - g a)*(f x)"
- from prems have "a < b" by simp
- moreover have "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?h x"
- proof -
- have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. f b - f a) x" by simp
- with gc have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. (f b - f a) * g x) x"
- by (auto intro: isCont_mult)
- moreover
- have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. g b - g a) x" by simp
- with fc have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. (g b - g a) * f x) x"
- by (auto intro: isCont_mult)
- ultimately show ?thesis
- by (fastsimp intro: isCont_diff)
- qed
- moreover
- have "\<forall>x. a < x \<and> x < b \<longrightarrow> ?h differentiable x"
- proof -
- have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. f b - f a) differentiable x" by (simp add: differentiable_const)
- 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)
- moreover
- have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. g b - g a) differentiable x" by (simp add: differentiable_const)
- 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)
- ultimately show ?thesis by (simp add: differentiable_diff)
- qed
- ultimately have "\<exists>l z. a < z \<and> z < b \<and> DERIV ?h z :> l \<and> ?h b - ?h a = (b - a) * l" by (rule MVT)
- then obtain l where ldef: "\<exists>z. a < z \<and> z < b \<and> DERIV ?h z :> l \<and> ?h b - ?h a = (b - a) * l" ..
- then obtain c where cdef: "a < c \<and> c < b \<and> DERIV ?h c :> l \<and> ?h b - ?h a = (b - a) * l" ..
-
- from cdef have cint: "a < c \<and> c < b" by auto
- with gd have "g differentiable c" by simp
- hence "\<exists>D. DERIV g c :> D" by (rule differentiableD)
- then obtain g'c where g'cdef: "DERIV g c :> g'c" ..
-
- from cdef have "a < c \<and> c < b" by auto
- with fd have "f differentiable c" by simp
- hence "\<exists>D. DERIV f c :> D" by (rule differentiableD)
- then obtain f'c where f'cdef: "DERIV f c :> f'c" ..
-
- from cdef have "DERIV ?h c :> l" by auto
- moreover
- {
- from g'cdef have "DERIV (\<lambda>x. (f b - f a) * g x) c :> g'c * (f b - f a)"
- apply (insert DERIV_const [where k="f b - f a"])
- apply (drule meta_spec [of _ c])
- apply (drule DERIV_mult [where f="(\<lambda>x. f b - f a)" and g=g])
- by simp_all
- moreover from f'cdef have "DERIV (\<lambda>x. (g b - g a) * f x) c :> f'c * (g b - g a)"
- apply (insert DERIV_const [where k="g b - g a"])
- apply (drule meta_spec [of _ c])
- apply (drule DERIV_mult [where f="(\<lambda>x. g b - g a)" and g=f])
- by simp_all
- ultimately have "DERIV ?h c :> g'c * (f b - f a) - f'c * (g b - g a)"
- by (simp add: DERIV_diff)
- }
- ultimately have leq: "l = g'c * (f b - f a) - f'c * (g b - g a)" by (rule DERIV_unique)
-
- {
- from cdef have "?h b - ?h a = (b - a) * l" by auto
- also with leq have "\<dots> = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
- finally have "?h b - ?h a = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
- }
- moreover
- {
- have "?h b - ?h a =
- ((f b)*(g b) - (f a)*(g b) - (g b)*(f b) + (g a)*(f b)) -
- ((f b)*(g a) - (f a)*(g a) - (g b)*(f a) + (g a)*(f a))"
- by (simp add: mult_ac add_ac real_diff_mult_distrib)
- hence "?h b - ?h a = 0" by auto
- }
- ultimately have "(b - a) * (g'c * (f b - f a) - f'c * (g b - g a)) = 0" by auto
- with alb have "g'c * (f b - f a) - f'c * (g b - g a) = 0" by simp
- hence "g'c * (f b - f a) = f'c * (g b - g a)" by simp
- hence "(f b - f a) * g'c = (g b - g a) * f'c" by (simp add: mult_ac)
-
- with g'cdef f'cdef cint show ?thesis by auto
-qed
-
+subsection {* Relation of LIM and LIMSEQ *}
lemma LIMSEQ_SEQ_conv1:
- fixes a :: real
- assumes "X -- a --> L"
+ fixes a :: "'a::real_normed_vector"
+ assumes X: "X -- a --> L"
shows "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
-proof -
- {
- from prems have Xdef: "\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & norm (x - a) < s --> norm (X x - L) < r" by (unfold LIM_def)
-
- fix S
- assume as: "(\<forall>n. S n \<noteq> a) \<and> S ----> a"
- then have "S ----> a" by auto
- then have Sdef: "(\<forall>r. 0 < r --> (\<exists>no. \<forall>n. no \<le> n --> norm (S n - a) < r))" by (unfold LIMSEQ_def)
- {
- fix r
- from Xdef have Xdef2: "0 < r --> (\<exists>s > 0. \<forall>x. x \<noteq> a \<and> \<bar>x - a\<bar> < s --> norm (X x - L) < r)" by simp
- {
- assume rgz: "0 < r"
-
- from Xdef2 rgz have "\<exists>s > 0. \<forall>x. x \<noteq> a \<and> \<bar>x - a\<bar> < s --> norm (X x - L) < r" by simp
- then obtain s where sdef: "s > 0 \<and> (\<forall>x. x\<noteq>a \<and> \<bar>x - a\<bar> < s \<longrightarrow> norm (X x - L) < r)" by auto
- then have aux: "\<forall>x. x\<noteq>a \<and> \<bar>x - a\<bar> < s \<longrightarrow> norm (X x - L) < r" by auto
- {
- fix n
- from aux have "S n \<noteq> a \<and> \<bar>S n - a\<bar> < s \<longrightarrow> norm (X (S n) - L) < r" by simp
- with as have imp2: "\<bar>S n - a\<bar> < s --> norm (X (S n) - L) < r" by auto
- }
- hence "\<forall>n. \<bar>S n - a\<bar> < s --> norm (X (S n) - L) < r" ..
- moreover
- from Sdef sdef have imp1: "\<exists>no. \<forall>n. no \<le> n --> \<bar>S n - a\<bar> < s" by auto
- then obtain no where "\<forall>n. no \<le> n --> \<bar>S n - a\<bar> < s" by auto
- ultimately have "\<forall>n. no \<le> n \<longrightarrow> norm (X (S n) - L) < r" by simp
- hence "\<exists>no. \<forall>n. no \<le> n \<longrightarrow> norm (X (S n) - L) < r" by auto
- }
- }
- hence "(\<forall>r. 0 < r --> (\<exists>no. \<forall>n. no \<le> n --> norm (X (S n) - L) < r))" by simp
- hence "(\<lambda>n. X (S n)) ----> L" by (fold LIMSEQ_def)
- }
- thus ?thesis by simp
+proof (safe intro!: LIMSEQ_I)
+ fix S :: "nat \<Rightarrow> 'a"
+ fix r :: real
+ assume rgz: "0 < r"
+ assume as: "\<forall>n. S n \<noteq> a"
+ assume S: "S ----> a"
+ from LIM_D [OF X rgz] obtain s
+ where sgz: "0 < s"
+ and aux: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < s\<rbrakk> \<Longrightarrow> norm (X x - L) < r"
+ by fast
+ from LIMSEQ_D [OF S sgz]
+ obtain no where "\<forall>n\<ge>no. norm (S n - a) < s" by fast
+ hence "\<forall>n\<ge>no. norm (X (S n) - L) < r" by (simp add: aux as)
+ thus "\<exists>no. \<forall>n\<ge>no. norm (X (S n) - L) < r" ..
qed
-ML {* fast_arith_split_limit := 0; *} (* FIXME *)
-
lemma LIMSEQ_SEQ_conv2:
fixes a :: real
assumes "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
@@ -2460,51 +705,39 @@
then obtain r where rdef: "r > 0 \<and> (\<forall>s > 0. \<exists>x. (x \<noteq> a \<and> \<bar>x - a\<bar> < s \<and> norm (X x - L) \<ge> r))" by auto
let ?F = "\<lambda>n::nat. SOME x. x\<noteq>a \<and> \<bar>x - a\<bar> < inverse (real (Suc n)) \<and> norm (X x - L) \<ge> r"
+ have "\<And>n. \<exists>x. x\<noteq>a \<and> \<bar>x - a\<bar> < inverse (real (Suc n)) \<and> norm (X x - L) \<ge> r"
+ using rdef by simp
+ hence F: "\<And>n. ?F n \<noteq> a \<and> \<bar>?F n - a\<bar> < inverse (real (Suc n)) \<and> norm (X (?F n) - L) \<ge> r"
+ by (rule someI_ex)
+ hence F1: "\<And>n. ?F n \<noteq> a"
+ and F2: "\<And>n. \<bar>?F n - a\<bar> < inverse (real (Suc n))"
+ and F3: "\<And>n. norm (X (?F n) - L) \<ge> r"
+ by fast+
+
have "?F ----> a"
- proof -
- {
+ proof (rule LIMSEQ_I, unfold real_norm_def)
fix e::real
assume "0 < e"
(* choose no such that inverse (real (Suc n)) < e *)
have "\<exists>no. inverse (real (Suc no)) < e" by (rule reals_Archimedean)
then obtain m where nodef: "inverse (real (Suc m)) < e" by auto
- {
+ show "\<exists>no. \<forall>n. no \<le> n --> \<bar>?F n - a\<bar> < e"
+ proof (intro exI allI impI)
fix n
assume mlen: "m \<le> n"
- then have
- "inverse (real (Suc n)) \<le> inverse (real (Suc m))"
+ have "\<bar>?F n - a\<bar> < inverse (real (Suc n))"
+ by (rule F2)
+ also have "inverse (real (Suc n)) \<le> inverse (real (Suc m))"
by auto
- moreover have
- "\<bar>?F n - a\<bar> < inverse (real (Suc n))"
- proof -
- from rdef have
- "\<exists>x. x\<noteq>a \<and> \<bar>x - a\<bar> < inverse (real (Suc n)) \<and> norm (X x - L) \<ge> r"
- by simp
- hence
- "(?F n)\<noteq>a \<and> \<bar>(?F n) - a\<bar> < inverse (real (Suc n)) \<and> norm (X (?F n) - L) \<ge> r"
- by (simp add: some_eq_ex [symmetric])
- thus ?thesis by simp
- qed
- moreover from nodef have
+ also from nodef have
"inverse (real (Suc m)) < e" .
- ultimately have "\<bar>?F n - a\<bar> < e" by arith
- }
- then have "\<exists>no. \<forall>n. no \<le> n --> \<bar>?F n - a\<bar> < e" by auto
- }
- thus ?thesis by (unfold LIMSEQ_def, simp)
+ finally show "\<bar>?F n - a\<bar> < e" .
+ qed
qed
moreover have "\<forall>n. ?F n \<noteq> a"
- proof -
- {
- fix n
- from rdef have
- "\<exists>x. x\<noteq>a \<and> \<bar>x - a\<bar> < inverse (real (Suc n)) \<and> norm (X x - L) \<ge> r"
- by simp
- hence "?F n \<noteq> a" by (simp add: some_eq_ex [symmetric])
- }
- thus ?thesis ..
- qed
+ by (rule allI) (rule F1)
+
moreover from prems have "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L" by simp
ultimately have "(\<lambda>n. X (?F n)) ----> L" by simp
@@ -2515,12 +748,9 @@
obtain n where "n = no + 1" by simp
then have nolen: "no \<le> n" by simp
(* We prove this by showing that for any m there is an n\<ge>m such that |X (?F n) - L| \<ge> r *)
- from rdef have "\<forall>s > 0. \<exists>x. (x \<noteq> a \<and> \<bar>x - a\<bar> < s \<and> norm (X x - L) \<ge> r)" ..
-
- then have "\<exists>x. x\<noteq>a \<and> \<bar>x - a\<bar> < inverse (real (Suc n)) \<and> norm (X x - L) \<ge> r" by simp
-
- hence "norm (X (?F n) - L) \<ge> r" by (simp add: some_eq_ex [symmetric])
- with nolen have "\<exists>n. no \<le> n \<and> norm (X (?F n) - L) \<ge> r" by auto
+ have "norm (X (?F n) - L) \<ge> r"
+ by (rule F3)
+ with nolen have "\<exists>n. no \<le> n \<and> norm (X (?F n) - L) \<ge> r" by fast
}
then have "(\<forall>no. \<exists>n. no \<le> n \<and> norm (X (?F n) - L) \<ge> r)" by simp
with rdef have "\<exists>e>0. (\<forall>no. \<exists>n. no \<le> n \<and> norm (X (?F n) - L) \<ge> e)" by auto
@@ -2529,8 +759,6 @@
ultimately show False by simp
qed
-ML {* fast_arith_split_limit := 9; *} (* FIXME *)
-
lemma LIMSEQ_SEQ_conv:
"(\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> (a::real) \<longrightarrow> (\<lambda>n. X (S n)) ----> L) =
(X -- a --> L)"
@@ -2542,39 +770,4 @@
show "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L" by (rule LIMSEQ_SEQ_conv1)
qed
-lemma real_sqz:
- fixes a::real
- assumes "a < c"
- shows "\<exists>b. a < b \<and> b < c"
-by (rule dense)
-
-lemma LIM_offset:
- assumes "(\<lambda>x. f x) -- a --> L"
- shows "(\<lambda>x. f (x+c)) -- (a-c) --> L"
-proof -
- have "f -- a --> L" .
- hence
- fd: "\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & norm (x - a) < s --> norm (f x - L) < r"
- by (unfold LIM_def)
- {
- fix r::real
- assume rgz: "0 < r"
- with fd have "\<exists>s > 0. \<forall>x. x\<noteq>a \<and> norm (x - a) < s --> norm (f x - L) < r" by simp
- then obtain s where sgz: "s > 0" and ax: "\<forall>x. x\<noteq>a \<and> norm (x - a) < s \<longrightarrow> norm (f x - L) < r" by auto
- from ax have ax2: "\<forall>x. (x+c)\<noteq>a \<and> norm ((x+c) - a) < s \<longrightarrow> norm (f (x+c) - L) < r" by auto
- {
- fix x
- from ax2 have nt: "(x+c)\<noteq>a \<and> norm ((x+c) - a) < s \<longrightarrow> norm (f (x+c) - L) < r" ..
- moreover have "((x+c)\<noteq>a) = (x\<noteq>(a-c))" by auto
- moreover have "((x+c) - a) = (x - (a-c))" by simp
- ultimately have "x\<noteq>(a-c) \<and> norm (x - (a-c)) < s \<longrightarrow> norm (f (x+c) - L) < r" by simp
- }
- then have "\<forall>x. x\<noteq>(a-c) \<and> norm (x - (a-c)) < s \<longrightarrow> norm (f (x+c) - L) < r" ..
- with sgz have "\<exists>s > 0. \<forall>x. x\<noteq>(a-c) \<and> norm (x - (a-c)) < s \<longrightarrow> norm (f (x+c) - L) < r" by auto
- }
- then have
- "\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> (a-c) & norm (x - (a-c)) < s --> norm (f (x+c) - L) < r" by simp
- thus ?thesis by (fold LIM_def)
-qed
-
end