author huffman Sat, 04 Nov 2006 00:12:06 +0100 changeset 21165 8fb49f668511 parent 21164 0742fc979c67 child 21166 2075d9027004
moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
 src/HOL/Hyperreal/Lim.thy file | annotate | diff | comparison | revisions src/HOL/Hyperreal/Transcendental.thy file | annotate | diff | comparison | revisions
```--- 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
*)

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"

-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)"
-
-lemma DERIV_D: "DERIV f x :> D ==> (%h. (f(x + h) - f(x))/#h) -- 0 --> D"
-
-lemma DERIV_const [simp]: "DERIV (\<lambda>x. k) x :> 0"
-
-lemma DERIV_Id [simp]: "DERIV (\<lambda>x. x) x :> 1"
-by (simp add: deriv_def real_scaleR_def cong: LIM_cong)
-
-  fixes a b c d :: "'a::ab_group_add"
-  shows "(a + c) - (b + d) = (a - b) + (c - d)"
-by simp
-
-  "\<lbrakk>DERIV f x :> D; DERIV g x :> E\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. f x + g x) x :> D + E"
-
-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)
-
-  "\<lbrakk>DERIV f x :> D; DERIV g x :> E\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. f x + - g x) x :> D + - E"
-
-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)"
-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 (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")
-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 (drule_tac k="a" in LIM_shift)
-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 (drule_tac k="a" in LIM_shift)
-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)"
-
-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)
-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')
-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)"
-
-lemma NS_DERIV_D: "DERIV f x :> D ==> (%h. (f(x + h) - f(x))/#h) -- 0 --NS> D"
-
-lemma NSDeriv_unique:
-     "[| NSDERIV f x :> D; NSDERIV f x :> E |] ==> D = E"
-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)"
-
-(*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 (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) ")
-         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 (case_tac "h = (0::hypreal) ")
-apply (drule_tac x = h in bspec)
-apply (drule_tac [2] c = h in approx_mult1)
-apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
-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 (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]
-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)
-apply (drule_tac c = "xa - hypreal_of_real x" in approx_mult1)
-apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
-apply (drule_tac x3=D in
-           HFinite_hypreal_of_real [THEN [2] Infinitesimal_HFinite_mult,
-             THEN mem_infmal_iff [THEN iffD1]])
-            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)"
-
-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 (drule_tac b = "hypreal_of_real Da" and d = "hypreal_of_real Db" in approx_add)
-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))"
-
-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)
-            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
-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 (rule_tac b1 = "hypreal_of_real Db * hypreal_of_real (f x)"
-apply (auto intro!: Infinitesimal_add_approx_self2 [THEN approx_sym]
-                    Infinitesimal_hypreal_of_real_mult
-                    Infinitesimal_hypreal_of_real_mult2
-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"
-  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"
-  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"
-
-lemma NSDERIV_diff:
-     "[| NSDERIV f x :> Da; NSDERIV g x :> Db |]
-      ==> NSDERIV (%x. f x - g x) x :> Da-Db"
-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 (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 (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 (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 (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 (rule ballI, simp, clarify)
-(*apply (auto simp add: starfun_inverse_inverse realpow_two
-        simp del: minus_mult_left [symmetric] minus_mult_right [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 (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])
-done
-
-(* NS version *)
-lemma NSDERIV_pow: "NSDERIV (%x. x ^ n) x :> real n * (x ^ (n - Suc 0))"
-
-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"
-
-lemma differentiableI: "DERIV f x :> D ==> f differentiable x"
-
-lemma NSdifferentiableD: "f NSdifferentiable x ==> \<exists>D. NSDERIV f x :> D"
-
-lemma NSdifferentiableI: "NSDERIV f x :> D ==> f NSdifferentiable x"
-
-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)"
-
-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
-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,
-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 (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])
-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")
-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 (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 (rule real_sum_of_halves [THEN subst])
-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 (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 (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 (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)
-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)"
-  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]
-      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)"
-  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]
-      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 (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],
-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)
-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"]
-  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"]
-  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])
-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)"
-  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)"
-  }
-  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))"
-    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```
```--- a/src/HOL/Hyperreal/Transcendental.thy	Sat Nov 04 00:11:11 2006 +0100
+++ b/src/HOL/Hyperreal/Transcendental.thy	Sat Nov 04 00:12:06 2006 +0100
@@ -8,7 +8,7 @@