--- a/src/HOL/Hyperreal/Lim.thy Sun Sep 17 16:42:38 2006 +0200
+++ b/src/HOL/Hyperreal/Lim.thy Sun Sep 17 16:44:05 2006 +0200
@@ -15,21 +15,21 @@
text{*Standard and Nonstandard Definitions*}
definition
- LIM :: "[real => 'a::real_normed_vector, real, 'a] => bool"
+ LIM :: "['a::real_normed_vector => 'b::real_normed_vector, 'a, 'b] => bool"
("((_)/ -- (_)/ --> (_))" [60, 0, 60] 60)
"f -- a --> L =
- (\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & \<bar>x + -a\<bar> < s
+ (\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & norm (x + -a) < s
--> norm (f x + -L) < r)"
- NSLIM :: "[real => 'a::real_normed_vector, real, 'a] => bool"
+ NSLIM :: "['a::real_normed_vector => 'b::real_normed_vector, 'a, 'b] => bool"
("((_)/ -- (_)/ --NS> (_))" [60, 0, 60] 60)
"f -- a --NS> L =
(\<forall>x. (x \<noteq> star_of a & x @= star_of a --> ( *f* f) x @= star_of L))"
- isCont :: "[real => 'a::real_normed_vector, real] => bool"
+ isCont :: "['a::real_normed_vector => 'b::real_normed_vector, 'a] => bool"
"isCont f a = (f -- a --> (f a))"
- isNSCont :: "[real => 'a::real_normed_vector, real] => bool"
+ isNSCont :: "['a::real_normed_vector => 'b::real_normed_vector, 'a] => bool"
--{*NS definition dispenses with limit notions*}
"isNSCont f a = (\<forall>y. y @= star_of a -->
( *f* f) y @= star_of (f a))"
@@ -78,23 +78,24 @@
lemma LIM_eq:
"f -- a --> L =
- (\<forall>r>0.\<exists>s>0.\<forall>x. x \<noteq> a & \<bar>x-a\<bar> < s --> norm (f x - L) < r)"
+ (\<forall>r>0.\<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)"
by (simp add: LIM_def diff_minus)
lemma LIM_I:
- "(!!r. 0<r ==> \<exists>s>0.\<forall>x. x \<noteq> a & \<bar>x-a\<bar> < s --> norm (f x - L) < r)
+ "(!!r. 0<r ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)
==> f -- a --> L"
by (simp add: LIM_eq)
lemma LIM_D:
"[| f -- a --> L; 0<r |]
- ==> \<exists>s>0.\<forall>x. x \<noteq> a & \<bar>x-a\<bar> < s --> norm (f x - L) < r"
+ ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r"
by (simp add: LIM_eq)
lemma LIM_const [simp]: "(%x. k) -- x --> k"
by (simp add: LIM_def)
lemma LIM_add:
+ fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
assumes f: "f -- a --> L" and g: "g -- a --> M"
shows "(%x. f x + g(x)) -- a --> (L + M)"
proof (rule LIM_I)
@@ -103,19 +104,19 @@
from LIM_D [OF f half_gt_zero [OF r]]
obtain fs
where fs: "0 < fs"
- and fs_lt: "\<forall>x. x \<noteq> a & \<bar>x-a\<bar> < fs --> norm (f x - L) < r/2"
+ and fs_lt: "\<forall>x. x \<noteq> a & norm (x-a) < fs --> norm (f x - L) < r/2"
by blast
from LIM_D [OF g half_gt_zero [OF r]]
obtain gs
where gs: "0 < gs"
- and gs_lt: "\<forall>x. x \<noteq> a & \<bar>x-a\<bar> < gs --> norm (g x - M) < r/2"
+ and gs_lt: "\<forall>x. x \<noteq> a & norm (x-a) < gs --> norm (g x - M) < r/2"
by blast
- show "\<exists>s>0.\<forall>x. x \<noteq> a \<and> \<bar>x-a\<bar> < s \<longrightarrow> norm (f x + g x - (L + M)) < r"
+ show "\<exists>s>0.\<forall>x. x \<noteq> a \<and> norm (x-a) < s \<longrightarrow> norm (f x + g x - (L + M)) < r"
proof (intro exI conjI strip)
show "0 < min fs gs" by (simp add: fs gs)
- fix x :: real
- assume "x \<noteq> a \<and> \<bar>x-a\<bar> < min fs gs"
- hence "x \<noteq> a \<and> \<bar>x-a\<bar> < fs \<and> \<bar>x-a\<bar> < gs" by simp
+ fix x :: 'a
+ assume "x \<noteq> a \<and> norm (x-a) < min fs gs"
+ hence "x \<noteq> a \<and> norm (x-a) < fs \<and> norm (x-a) < gs" by simp
with fs_lt gs_lt
have "norm (f x - L) < r/2" and "norm (g x - M) < r/2" by blast+
hence "norm (f x - L) + norm (g x - M) < r" by arith
@@ -141,24 +142,30 @@
by (simp only: diff_minus LIM_add LIM_minus)
-lemma LIM_const_not_eq: "k \<noteq> L ==> ~ ((%x. k) -- a --> L)"
+lemma LIM_const_not_eq:
+ fixes a :: "'a::real_normed_div_algebra"
+ shows "k \<noteq> L ==> ~ ((%x. k) -- a --> L)"
apply (simp add: LIM_eq)
apply (rule_tac x="norm (k - L)" in exI, simp, safe)
-apply (rule_tac x="a + s/2" in exI, simp)
+apply (rule_tac x="a + of_real (s/2)" in exI, simp add: norm_of_real)
done
-lemma LIM_const_eq: "(%x. k) -- x --> L ==> k = L"
+lemma LIM_const_eq:
+ fixes a :: "'a::real_normed_div_algebra"
+ shows "(%x. k) -- a --> L ==> k = L"
apply (rule ccontr)
apply (blast dest: LIM_const_not_eq)
done
-lemma LIM_unique: "[| f -- a --> L; f -- a --> M |] ==> L = M"
+lemma LIM_unique:
+ fixes a :: "'a::real_normed_div_algebra"
+ shows "[| f -- a --> L; f -- a --> M |] ==> L = M"
apply (drule LIM_diff, assumption)
apply (auto dest!: LIM_const_eq)
done
lemma LIM_mult_zero:
- fixes f g :: "real \<Rightarrow> 'a::real_normed_algebra"
+ fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
assumes f: "f -- a --> 0" and g: "g -- a --> 0"
shows "(%x. f(x) * g(x)) -- a --> 0"
proof (rule LIM_I, simp)
@@ -167,19 +174,19 @@
from LIM_D [OF f zero_less_one]
obtain fs
where fs: "0 < fs"
- and fs_lt: "\<forall>x. x \<noteq> a & \<bar>x-a\<bar> < fs --> norm (f x) < 1"
+ and fs_lt: "\<forall>x. x \<noteq> a & norm (x-a) < fs --> norm (f x) < 1"
by auto
from LIM_D [OF g r]
obtain gs
where gs: "0 < gs"
- and gs_lt: "\<forall>x. x \<noteq> a & \<bar>x-a\<bar> < gs --> norm (g x) < r"
+ and gs_lt: "\<forall>x. x \<noteq> a & norm (x-a) < gs --> norm (g x) < r"
by auto
- show "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> a \<and> \<bar>x-a\<bar> < s \<longrightarrow> norm (f x * g x) < r)"
+ show "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> a \<and> norm (x-a) < s \<longrightarrow> norm (f x * g x) < r)"
proof (intro exI conjI strip)
show "0 < min fs gs" by (simp add: fs gs)
- fix x :: real
- assume "x \<noteq> a \<and> \<bar>x-a\<bar> < min fs gs"
- hence "x \<noteq> a \<and> \<bar>x-a\<bar> < fs \<and> \<bar>x-a\<bar> < gs" by simp
+ fix x :: 'a
+ assume "x \<noteq> a \<and> norm (x-a) < min fs gs"
+ hence "x \<noteq> a \<and> norm (x-a) < fs \<and> norm (x-a) < gs" by simp
with fs_lt gs_lt
have "norm (f x) < 1" and "norm (g x) < r" by blast+
hence "norm (f x) * norm (g x) < 1*r"
@@ -226,18 +233,18 @@
lemma lemma_LIM:
fixes L :: "'a::real_normed_vector"
- shows "\<forall>s>0. \<exists>x. x \<noteq> a \<and> \<bar>x + - a\<bar> < s \<and> \<not> norm (f x + -L) < r
+ shows "\<forall>s>0. \<exists>x. x \<noteq> a \<and> norm (x + - a) < s \<and> \<not> norm (f x + -L) < r
==> \<forall>n::nat. \<exists>x. x \<noteq> a \<and>
- \<bar>x + -a\<bar> < inverse(real(Suc n)) \<and> \<not> norm (f x + -L) < r"
+ norm (x + -a) < inverse(real(Suc n)) \<and> \<not> norm (f x + -L) < r"
apply clarify
apply (cut_tac n1 = n in real_of_nat_Suc_gt_zero [THEN positive_imp_inverse_positive], auto)
done
lemma lemma_skolemize_LIM2:
fixes L :: "'a::real_normed_vector"
- shows "\<forall>s>0. \<exists>x. x \<noteq> a \<and> \<bar>x + - a\<bar> < s \<and> \<not> norm (f x + -L) < r
+ shows "\<forall>s>0. \<exists>x. x \<noteq> a \<and> norm (x + - a) < s \<and> \<not> norm (f x + -L) < r
==> \<exists>X. \<forall>n::nat. X n \<noteq> a \<and>
- \<bar>X n + -a\<bar> < inverse(real(Suc n)) \<and> \<not> norm(f (X n) + -L) < r"
+ norm (X n + -a) < inverse(real(Suc n)) \<and> \<not> norm(f (X n) + -L) < r"
apply (drule lemma_LIM)
apply (drule choice, blast)
done
@@ -252,9 +259,9 @@
done
lemma lemma_simp: "\<forall>n. X n \<noteq> a &
- \<bar>X n + - a\<bar> < inverse (real(Suc n)) &
+ norm (X n + - a) < inverse (real(Suc n)) &
\<not> norm (f (X n) + - L) < r ==>
- \<forall>n. \<bar>X n + - a\<bar> < inverse (real(Suc n))"
+ \<forall>n. norm (X n + - a) < inverse (real(Suc n))"
by auto
@@ -359,35 +366,45 @@
apply (auto simp add: diff_minus add_assoc)
done
-lemma NSLIM_const_not_eq: "k \<noteq> L ==> ~ ((%x. k) -- x --NS> L)"
+lemma NSLIM_const_not_eq:
+ fixes a :: real (* TODO: generalize to real_normed_div_algebra *)
+ shows "k \<noteq> L ==> ~ ((%x. k) -- a --NS> L)"
apply (simp add: NSLIM_def)
-apply (rule_tac x = "hypreal_of_real x + epsilon" in exI)
+apply (rule_tac x="star_of a + epsilon" in exI)
apply (auto intro: Infinitesimal_add_approx_self [THEN approx_sym]
simp add: hypreal_epsilon_not_zero)
done
-lemma NSLIM_not_zero: "k \<noteq> 0 ==> ~ ((%x. k) -- x --NS> 0)"
+lemma NSLIM_not_zero:
+ fixes a :: real
+ shows "k \<noteq> 0 ==> ~ ((%x. k) -- a --NS> 0)"
by (rule NSLIM_const_not_eq)
-lemma NSLIM_const_eq: "(%x. k) -- x --NS> L ==> k = L"
+lemma NSLIM_const_eq:
+ fixes a :: real
+ shows "(%x. k) -- a --NS> L ==> k = L"
apply (rule ccontr)
apply (blast dest: NSLIM_const_not_eq)
done
text{* can actually be proved more easily by unfolding the definition!*}
-lemma NSLIM_unique: "[| f -- x --NS> L; f -- x --NS> M |] ==> L = M"
+lemma NSLIM_unique:
+ fixes a :: real
+ shows "[| f -- a --NS> L; f -- a --NS> M |] ==> L = M"
apply (drule NSLIM_minus)
apply (drule NSLIM_add, assumption)
apply (auto dest!: NSLIM_const_eq [symmetric])
apply (simp add: diff_def [symmetric])
done
-lemma LIM_unique2: "[| f -- x --> L; f -- x --> M |] ==> L = M"
+lemma LIM_unique2:
+ fixes a :: real
+ shows "[| f -- a --> L; f -- a --> M |] ==> L = M"
by (simp add: LIM_NSLIM_iff NSLIM_unique)
lemma NSLIM_mult_zero:
- fixes f g :: "real \<Rightarrow> 'a::real_normed_algebra"
+ fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
shows "[| f -- x --NS> 0; g -- x --NS> 0 |] ==> (%x. f(x)*g(x)) -- x --NS> 0"
by (drule NSLIM_mult, auto)
@@ -395,7 +412,7 @@
(* for standard definition of limit *)
lemma LIM_mult_zero2:
- fixes f g :: "real \<Rightarrow> 'a::real_normed_algebra"
+ fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
shows "[| f -- x --> 0; g -- x --> 0 |] ==> (%x. f(x)*g(x)) -- x --> 0"
by (drule LIM_mult2, auto)
@@ -416,7 +433,7 @@
lemma NSLIM_isNSCont: "f -- a --NS> (f a) ==> isNSCont f a"
apply (simp add: isNSCont_def NSLIM_def, auto)
-apply (rule_tac Q = "y = hypreal_of_real a" in excluded_middle [THEN disjE], auto)
+apply (case_tac "y = star_of a", auto)
done
text{*NS continuity can be defined using NS Limit in
@@ -449,13 +466,13 @@
(* seems easier than using standard def *)
lemma NSLIM_h_iff: "(f -- a --NS> L) = ((%h. f(a + h)) -- 0 --NS> L)"
apply (simp add: NSLIM_def, auto)
-apply (drule_tac x = "hypreal_of_real a + x" in spec)
-apply (drule_tac [2] x = "-hypreal_of_real a + x" in spec, safe, simp)
-apply (rule mem_infmal_iff [THEN iffD2, THEN Infinitesimal_add_approx_self [THEN approx_sym]])
-apply (rule_tac [4] approx_minus_iff2 [THEN iffD1])
- prefer 3 apply (simp add: add_commute)
+apply (drule_tac x = "star_of a + x" in spec)
+apply (drule_tac [2] x = "- star_of a + x" in spec, safe, simp)
+apply (erule mem_infmal_iff [THEN iffD2, THEN Infinitesimal_add_approx_self [THEN approx_sym]])
+apply (erule_tac [3] approx_minus_iff2 [THEN iffD1])
+ prefer 2 apply (simp add: add_commute diff_def [symmetric])
+apply (rule_tac x = x in star_cases)
apply (rule_tac [2] x = x in star_cases)
-apply (rule_tac [4] x = x in star_cases)
apply (auto simp add: starfun star_of_def star_n_minus star_n_add add_assoc approx_refl star_n_zero_num)
done
@@ -476,7 +493,7 @@
text{*mult continuous*}
lemma isCont_mult:
- fixes f g :: "real \<Rightarrow> 'a::real_normed_algebra"
+ fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
shows "[| isCont f a; isCont g a |] ==> isCont (%x. f(x) * g(x)) a"
by (auto intro!: starfun_mult_HFinite_approx
simp del: starfun_mult [symmetric]
@@ -497,14 +514,16 @@
by (auto simp add: isNSCont_isCont_iff [symmetric] isNSCont_minus)
lemma isCont_inverse:
- fixes f :: "real \<Rightarrow> 'a::{real_normed_div_algebra,division_by_zero}"
+ fixes f :: "'a::real_normed_vector \<Rightarrow>
+ 'b::{real_normed_div_algebra,division_by_zero}"
shows "[| isCont f x; f x \<noteq> 0 |] ==> isCont (%x. inverse (f x)) x"
apply (simp add: isCont_def)
apply (blast intro: LIM_inverse)
done
lemma isNSCont_inverse:
- fixes f :: "real \<Rightarrow> 'a::{real_normed_div_algebra,division_by_zero}"
+ fixes f :: "'a::real_normed_vector \<Rightarrow>
+ 'b::{real_normed_div_algebra,division_by_zero}"
shows "[| isNSCont f x; f x \<noteq> 0 |] ==> isNSCont (%x. inverse (f x)) x"
by (auto intro: isCont_inverse simp add: isNSCont_isCont_iff)
@@ -520,12 +539,12 @@
lemma isNSCont_const [simp]: "isNSCont (%x. k) a"
by (simp add: isNSCont_def)
-lemma isNSCont_abs [simp]: "isNSCont abs a"
+lemma isNSCont_abs [simp]: "isNSCont abs (a::real)"
apply (simp add: isNSCont_def)
apply (auto intro: approx_hrabs simp add: hypreal_of_real_hrabs [symmetric] starfun_rabs_hrabs)
done
-lemma isCont_abs [simp]: "isCont abs a"
+lemma isCont_abs [simp]: "isCont abs (a::real)"
by (auto simp add: isNSCont_isCont_iff [symmetric])
@@ -684,7 +703,7 @@
subsubsection{*Alternative definition for differentiability*}
lemma DERIV_LIM_iff:
- "((%h. (f(a + h) - f(a)) / h) -- 0 --> D) =
+ "((%h::real. (f(a + h) - f(a)) / h) -- 0 --> D) =
((%x. (f(x)-f(a)) / (x-a)) -- a --> D)"
proof (intro iffI LIM_I)
fix r::real
@@ -696,12 +715,12 @@
and s_lt: "\<forall>x. x \<noteq> 0 & \<bar>x\<bar> < s --> \<bar>(f (a + x) - f a) / x - D\<bar> < r"
by auto
show "\<exists>s. 0 < s \<and>
- (\<forall>x. x \<noteq> a \<and> \<bar>x-a\<bar> < s \<longrightarrow> norm ((f x - f a) / (x-a) - D) < r)"
+ (\<forall>x. x \<noteq> a \<and> norm (x-a) < s \<longrightarrow> norm ((f x - f a) / (x-a) - D) < r)"
proof (intro exI conjI strip)
show "0 < s" by (rule s)
next
fix x::real
- assume "x \<noteq> a \<and> \<bar>x-a\<bar> < s"
+ assume "x \<noteq> a \<and> norm (x-a) < s"
with s_lt [THEN spec [where x="x-a"]]
show "norm ((f x - f a) / (x-a) - D) < r" by auto
qed
@@ -715,12 +734,12 @@
and s_lt: "\<forall>x. x \<noteq> a & \<bar>x-a\<bar> < s --> \<bar>(f x - f a)/(x-a) - D\<bar> < r"
by auto
show "\<exists>s. 0 < s \<and>
- (\<forall>x. x \<noteq> 0 & \<bar>x - 0\<bar> < s --> norm ((f (a + x) - f a) / x - D) < r)"
+ (\<forall>x. x \<noteq> 0 & norm (x - 0) < s --> norm ((f (a + x) - f a) / x - D) < r)"
proof (intro exI conjI strip)
show "0 < s" by (rule s)
next
fix x::real
- assume "x \<noteq> 0 \<and> \<bar>x - 0\<bar> < s"
+ assume "x \<noteq> 0 \<and> norm (x - 0) < s"
with s_lt [THEN spec [where x="x+a"]]
show "norm ((f (a + x) - f a) / x - D) < r" by (auto simp add: add_ac)
qed
@@ -1415,7 +1434,7 @@
subsection{*Intermediate Value Theorem: Prove Contrapositive by Bisection*}
-lemma IVT: "[| f(a) \<le> (y::real); y \<le> f(b);
+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"
@@ -1442,7 +1461,7 @@
apply (case_tac "x \<le> aa", simp_all)
done
-lemma IVT2: "[| f(b) \<le> (y::real); y \<le> f(a);
+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"
@@ -1452,13 +1471,13 @@
done
(*HOL style here: object-level formulations*)
-lemma IVT_objl: "(f(a) \<le> (y::real) & y \<le> f(b) & a \<le> b &
+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) \<le> (y::real) & y \<le> f(a) & a \<le> b &
+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)
@@ -1468,7 +1487,7 @@
lemma isCont_bounded:
"[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
- ==> \<exists>M::real. \<forall>x. a \<le> x & x \<le> b --> f(x) \<le> M"
+ ==> \<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
@@ -1498,7 +1517,7 @@
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. a \<le> x & x \<le> b --> f(x) \<le> M) &
+ ==> \<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)
@@ -1513,7 +1532,7 @@
lemma isCont_eq_Ub:
assumes le: "a \<le> b"
- and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
+ 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 -
@@ -1554,7 +1573,7 @@
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. a \<le> x & x \<le> b --> M \<le> 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)
@@ -1567,7 +1586,7 @@
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. a \<le> x & x \<le> b --> L \<le> f(x) & f(x) \<le> M) &
+ ==> \<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)
@@ -2032,6 +2051,7 @@
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"
@@ -2205,11 +2225,12 @@
lemma LIMSEQ_SEQ_conv1:
+ fixes a :: real
assumes "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 & \<bar>x + -a\<bar> < s --> norm (X x + -L) < r" by (unfold LIM_def)
+ 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"
@@ -2246,11 +2267,12 @@
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"
shows "X -- a --> L"
proof (rule ccontr)
assume "\<not> (X -- a --> L)"
- hence "\<not> (\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & \<bar>x + -a\<bar> < s --> norm (X x + -L) < r)" by (unfold LIM_def)
+ hence "\<not> (\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & norm (x + -a) < s --> norm (X x + -L) < r)" by (unfold LIM_def)
hence "\<exists>r > 0. \<forall>s > 0. \<exists>x. \<not>(x \<noteq> a \<and> \<bar>x + -a\<bar> < s --> norm (X x + -L) < r)" by simp
hence "\<exists>r > 0. \<forall>s > 0. \<exists>x. (x \<noteq> a \<and> \<bar>x + -a\<bar> < s \<and> norm (X x + -L) \<ge> r)" by (simp add: linorder_not_less)
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
@@ -2328,7 +2350,8 @@
ML {* fast_arith_split_limit := 9; *} (* FIXME *)
lemma LIMSEQ_SEQ_conv:
- "(\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L) = (X -- a --> L)"
+ "(\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> (a::real) \<longrightarrow> (\<lambda>n. X (S n)) ----> L) =
+ (X -- a --> L)"
proof
assume "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
show "X -- a --> L" by (rule LIMSEQ_SEQ_conv2)
@@ -2356,26 +2379,26 @@
proof -
have "f -- a --> L" .
hence
- fd: "\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & \<bar>x + -a\<bar> < s --> norm (f x + -L) < r"
+ 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> \<bar>x + -a\<bar> < s --> norm (f x + -L) < r" by simp
- then obtain s where sgz: "s > 0" and ax: "\<forall>x. x\<noteq>a \<and> \<bar>x + -a\<bar> < s \<longrightarrow> norm (f x + -L) < r" by auto
- from ax have ax2: "\<forall>x. (x+c)\<noteq>a \<and> \<bar>(x+c) + -a\<bar> < s \<longrightarrow> norm (f (x+c) + -L) < r" by auto
+ 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::real
- from ax2 have nt: "(x+c)\<noteq>a \<and> \<bar>(x+c) + -a\<bar> < s \<longrightarrow> norm (f (x+c) + -L) < r" ..
+ 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> \<bar>x + -(a-c)\<bar> < s \<longrightarrow> norm (f (x+c) + -L) < r" 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> \<bar>x + -(a-c)\<bar> < s \<longrightarrow> norm (f (x+c) + -L) < r" ..
- with sgz have "\<exists>s > 0. \<forall>x. x\<noteq>(a-c) \<and> \<bar>x + -(a-c)\<bar> < s \<longrightarrow> norm (f (x+c) + -L) < r" by auto
+ 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) & \<bar>x + -(a-c)\<bar> < s --> norm (f (x+c) + -L) < r" by simp
+ "\<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
--- a/src/HOL/Hyperreal/Transcendental.thy Sun Sep 17 16:42:38 2006 +0200
+++ b/src/HOL/Hyperreal/Transcendental.thy Sun Sep 17 16:44:05 2006 +0200
@@ -530,7 +530,7 @@
done
lemma lemma_termdiff4:
- "[| 0 < k;
+ "[| 0 < (k::real);
(\<forall>h. 0 < \<bar>h\<bar> & \<bar>h\<bar> < k --> \<bar>f h\<bar> \<le> K * \<bar>h\<bar>) |]
==> f -- 0 --> 0"
apply (simp add: LIM_def, auto)
@@ -2520,7 +2520,8 @@
(* need to rename second isCont_inverse *)
lemma isCont_inv_fun:
- "[| 0 < d; \<forall>z. \<bar>z - x\<bar> \<le> d --> g(f(z)) = z;
+ 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 |]
==> isCont g (f x)"
apply (simp (no_asm) add: isCont_iff LIM_def)
@@ -2557,7 +2558,7 @@
text{*Bartle/Sherbert: Introduction to Real Analysis, Theorem 4.2.9, p. 110*}
lemma LIM_fun_gt_zero:
"[| f -- c --> (l::real); 0 < l |]
- ==> \<exists>r. 0 < r & (\<forall>x. x \<noteq> c & \<bar>c - x\<bar> < r --> 0 < f x)"
+ ==> \<exists>r. 0 < r & (\<forall>x::real. x \<noteq> c & \<bar>c - x\<bar> < r --> 0 < f x)"
apply (auto simp add: LIM_def)
apply (drule_tac x = "l/2" in spec, safe, force)
apply (rule_tac x = s in exI)
@@ -2566,7 +2567,7 @@
lemma LIM_fun_less_zero:
"[| f -- c --> (l::real); l < 0 |]
- ==> \<exists>r. 0 < r & (\<forall>x. x \<noteq> c & \<bar>c - x\<bar> < r --> f x < 0)"
+ ==> \<exists>r. 0 < r & (\<forall>x::real. x \<noteq> c & \<bar>c - x\<bar> < r --> f x < 0)"
apply (auto simp add: LIM_def)
apply (drule_tac x = "-l/2" in spec, safe, force)
apply (rule_tac x = s in exI)
@@ -2576,7 +2577,7 @@
lemma LIM_fun_not_zero:
"[| f -- c --> (l::real); l \<noteq> 0 |]
- ==> \<exists>r. 0 < r & (\<forall>x. x \<noteq> c & \<bar>c - x\<bar> < r --> f x \<noteq> 0)"
+ ==> \<exists>r. 0 < r & (\<forall>x::real. x \<noteq> c & \<bar>c - x\<bar> < r --> f x \<noteq> 0)"
apply (cut_tac x = l and y = 0 in linorder_less_linear, auto)
apply (drule LIM_fun_less_zero)
apply (drule_tac [3] LIM_fun_gt_zero)