--- a/src/HOL/Deriv.thy Fri Aug 19 15:07:10 2011 -0700
+++ b/src/HOL/Deriv.thy Fri Aug 19 15:54:43 2011 -0700
@@ -524,7 +524,7 @@
((\<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 (drule_tac [2] f = f in LIMSEQ_diff)
apply (auto intro: LIMSEQ_unique)
done
--- a/src/HOL/Lim.thy Fri Aug 19 15:07:10 2011 -0700
+++ b/src/HOL/Lim.thy Fri Aug 19 15:54:43 2011 -0700
@@ -81,32 +81,8 @@
shows "(\<lambda>h. f (a + h)) -- 0 --> L \<Longrightarrow> f -- a --> L"
by (drule_tac k="- a" in LIM_offset, simp)
-lemma LIM_const [simp]: "(%x. k) -- x --> k"
-by (rule tendsto_const)
-
lemma LIM_cong_limit: "\<lbrakk> f -- x --> L ; K = L \<rbrakk> \<Longrightarrow> f -- x --> K" by simp
-lemma LIM_add:
- fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
- assumes f: "f -- a --> L" and g: "g -- a --> M"
- shows "(\<lambda>x. f x + g x) -- a --> (L + M)"
-using assms by (rule tendsto_add)
-
-lemma LIM_add_zero:
- fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
- shows "\<lbrakk>f -- a --> 0; g -- a --> 0\<rbrakk> \<Longrightarrow> (\<lambda>x. f x + g x) -- a --> 0"
- by (rule tendsto_add_zero)
-
-lemma LIM_minus:
- fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
- shows "f -- a --> L \<Longrightarrow> (\<lambda>x. - f x) -- a --> - L"
-by (rule tendsto_minus)
-
-lemma LIM_diff:
- fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
- shows "\<lbrakk>f -- x --> l; g -- x --> m\<rbrakk> \<Longrightarrow> (\<lambda>x. f x - g x) -- x --> l - m"
-by (rule tendsto_diff)
-
lemma LIM_zero:
fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
shows "f -- a --> l \<Longrightarrow> (\<lambda>x. f x - l) -- a --> 0"
@@ -138,38 +114,6 @@
by (rule metric_LIM_imp_LIM [OF f],
simp add: dist_norm le)
-lemma LIM_norm:
- fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
- shows "f -- a --> l \<Longrightarrow> (\<lambda>x. norm (f x)) -- a --> norm l"
-by (rule tendsto_norm)
-
-lemma LIM_norm_zero:
- fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
- shows "f -- a --> 0 \<Longrightarrow> (\<lambda>x. norm (f x)) -- a --> 0"
-by (rule tendsto_norm_zero)
-
-lemma LIM_norm_zero_cancel:
- fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
- shows "(\<lambda>x. norm (f x)) -- a --> 0 \<Longrightarrow> f -- a --> 0"
-by (rule tendsto_norm_zero_cancel)
-
-lemma LIM_norm_zero_iff:
- fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
- shows "(\<lambda>x. norm (f x)) -- a --> 0 = f -- a --> 0"
-by (rule tendsto_norm_zero_iff)
-
-lemma LIM_rabs: "f -- a --> (l::real) \<Longrightarrow> (\<lambda>x. \<bar>f x\<bar>) -- a --> \<bar>l\<bar>"
- by (rule tendsto_rabs)
-
-lemma LIM_rabs_zero: "f -- a --> (0::real) \<Longrightarrow> (\<lambda>x. \<bar>f x\<bar>) -- a --> 0"
- by (rule tendsto_rabs_zero)
-
-lemma LIM_rabs_zero_cancel: "(\<lambda>x. \<bar>f x\<bar>) -- a --> (0::real) \<Longrightarrow> f -- a --> 0"
- by (rule tendsto_rabs_zero_cancel)
-
-lemma LIM_rabs_zero_iff: "(\<lambda>x. \<bar>f x\<bar>) -- a --> (0::real) = f -- a --> 0"
- by (rule tendsto_rabs_zero_iff)
-
lemma trivial_limit_at:
fixes a :: "'a::real_normed_algebra_1"
shows "\<not> trivial_limit (at a)" -- {* TODO: find a more appropriate class *}
@@ -197,9 +141,6 @@
shows "\<lbrakk>f -- a --> L; f -- a --> M\<rbrakk> \<Longrightarrow> L = M"
using trivial_limit_at by (rule tendsto_unique)
-lemma LIM_ident [simp]: "(\<lambda>x. x) -- a --> a"
-by (rule tendsto_ident_at)
-
text{*Limits are equal for functions equal except at limit point*}
lemma LIM_equal:
"[| \<forall>x. x \<noteq> a --> (f x = g x) |] ==> (f -- a --> l) = (g -- a --> l)"
@@ -229,12 +170,6 @@
shows "g -- a --> l \<Longrightarrow> f -- a --> l"
by (rule metric_LIM_equal2 [OF 1 2], simp_all add: dist_norm)
-lemma LIM_compose:
- assumes g: "g -- l --> g l"
- assumes f: "f -- a --> l"
- shows "(\<lambda>x. g (f x)) -- a --> g l"
- using assms by (rule tendsto_compose)
-
lemma LIM_compose_eventually:
assumes f: "f -- a --> b"
assumes g: "g -- b --> c"
@@ -247,8 +182,8 @@
assumes g: "g -- b --> c"
assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> b"
shows "(\<lambda>x. g (f x)) -- a --> c"
-using f g inj [folded eventually_at]
-by (rule LIM_compose_eventually)
+ using g f inj [folded eventually_at]
+ by (rule tendsto_compose_eventually)
lemma LIM_compose2:
fixes a :: "'a::real_normed_vector"
@@ -259,7 +194,7 @@
by (rule metric_LIM_compose2 [OF f g inj [folded dist_norm]])
lemma LIM_o: "\<lbrakk>g -- l --> g l; f -- a --> l\<rbrakk> \<Longrightarrow> (g \<circ> f) -- a --> g l"
-unfolding o_def by (rule LIM_compose)
+ unfolding o_def by (rule tendsto_compose)
lemma real_LIM_sandwich_zero:
fixes f g :: "'a::topological_space \<Rightarrow> real"
@@ -307,9 +242,6 @@
"f -- a --> 0 \<Longrightarrow> (\<lambda>x. c ** f x) -- a --> 0"
by (rule tendsto_right_zero)
-lemmas LIM_mult =
- bounded_bilinear.LIM [OF bounded_bilinear_mult]
-
lemmas LIM_mult_zero =
bounded_bilinear.LIM_prod_zero [OF bounded_bilinear_mult]
@@ -319,32 +251,10 @@
lemmas LIM_mult_right_zero =
bounded_bilinear.LIM_right_zero [OF bounded_bilinear_mult]
-lemmas LIM_scaleR =
- bounded_bilinear.LIM [OF bounded_bilinear_scaleR]
-
-lemmas LIM_of_real =
- bounded_linear.LIM [OF bounded_linear_of_real]
-
-lemma LIM_power:
- fixes f :: "'a::topological_space \<Rightarrow> 'b::{power,real_normed_algebra}"
- assumes f: "f -- a --> l"
- shows "(\<lambda>x. f x ^ n) -- a --> l ^ n"
- using assms by (rule tendsto_power)
-
-lemma LIM_inverse:
- fixes L :: "'a::real_normed_div_algebra"
- shows "\<lbrakk>f -- a --> L; L \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>x. inverse (f x)) -- a --> inverse L"
-by (rule tendsto_inverse)
-
lemma LIM_inverse_fun:
assumes a: "a \<noteq> (0::'a::real_normed_div_algebra)"
shows "inverse -- a --> inverse a"
-by (rule LIM_inverse [OF LIM_ident a])
-
-lemma LIM_sgn:
- fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
- shows "\<lbrakk>f -- a --> l; l \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>x. sgn (f x)) -- a --> sgn l"
- by (rule tendsto_sgn)
+ by (rule tendsto_inverse [OF tendsto_ident_at a])
subsection {* Continuity *}
@@ -360,45 +270,45 @@
by (simp add: isCont_def LIM_isCont_iff)
lemma isCont_ident [simp]: "isCont (\<lambda>x. x) a"
- unfolding isCont_def by (rule LIM_ident)
+ unfolding isCont_def by (rule tendsto_ident_at)
lemma isCont_const [simp]: "isCont (\<lambda>x. k) a"
- unfolding isCont_def by (rule LIM_const)
+ unfolding isCont_def by (rule tendsto_const)
lemma isCont_norm [simp]:
fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. norm (f x)) a"
- unfolding isCont_def by (rule LIM_norm)
+ unfolding isCont_def by (rule tendsto_norm)
lemma isCont_rabs [simp]:
fixes f :: "'a::topological_space \<Rightarrow> real"
shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. \<bar>f x\<bar>) a"
- unfolding isCont_def by (rule LIM_rabs)
+ unfolding isCont_def by (rule tendsto_rabs)
lemma isCont_add [simp]:
fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x + g x) a"
- unfolding isCont_def by (rule LIM_add)
+ unfolding isCont_def by (rule tendsto_add)
lemma isCont_minus [simp]:
fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. - f x) a"
- unfolding isCont_def by (rule LIM_minus)
+ unfolding isCont_def by (rule tendsto_minus)
lemma isCont_diff [simp]:
fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x - g x) a"
- unfolding isCont_def by (rule LIM_diff)
+ unfolding isCont_def by (rule tendsto_diff)
lemma isCont_mult [simp]:
fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_algebra"
shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x * g x) a"
- unfolding isCont_def by (rule LIM_mult)
+ unfolding isCont_def by (rule tendsto_mult)
lemma isCont_inverse [simp]:
fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_div_algebra"
shows "\<lbrakk>isCont f a; f a \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. inverse (f x)) a"
- unfolding isCont_def by (rule LIM_inverse)
+ unfolding isCont_def by (rule tendsto_inverse)
lemma isCont_divide [simp]:
fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_field"
@@ -409,10 +319,6 @@
"\<lbrakk>isCont g l; (f ---> l) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. g (f x)) ---> g l) F"
unfolding isCont_def by (rule tendsto_compose)
-lemma isCont_LIM_compose:
- "\<lbrakk>isCont g l; f -- a --> l\<rbrakk> \<Longrightarrow> (\<lambda>x. g (f x)) -- a --> g l"
- by (rule isCont_tendsto_compose) (* TODO: delete? *)
-
lemma metric_isCont_LIM_compose2:
assumes f [unfolded isCont_def]: "isCont f a"
assumes g: "g -- f a --> l"
@@ -429,18 +335,18 @@
by (rule LIM_compose2 [OF f g inj])
lemma isCont_o2: "\<lbrakk>isCont f a; isCont g (f a)\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. g (f x)) a"
- unfolding isCont_def by (rule LIM_compose)
+ unfolding isCont_def by (rule tendsto_compose)
lemma isCont_o: "\<lbrakk>isCont f a; isCont g (f a)\<rbrakk> \<Longrightarrow> isCont (g o f) a"
unfolding o_def by (rule isCont_o2)
lemma (in bounded_linear) isCont:
"isCont g a \<Longrightarrow> isCont (\<lambda>x. f (g x)) a"
- unfolding isCont_def by (rule LIM)
+ unfolding isCont_def by (rule tendsto)
lemma (in bounded_bilinear) isCont:
"\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x ** g x) a"
- unfolding isCont_def by (rule LIM)
+ unfolding isCont_def by (rule tendsto)
lemmas isCont_scaleR [simp] =
bounded_bilinear.isCont [OF bounded_bilinear_scaleR]
@@ -451,12 +357,12 @@
lemma isCont_power [simp]:
fixes f :: "'a::topological_space \<Rightarrow> 'b::{power,real_normed_algebra}"
shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x ^ n) a"
- unfolding isCont_def by (rule LIM_power)
+ unfolding isCont_def by (rule tendsto_power)
lemma isCont_sgn [simp]:
fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
shows "\<lbrakk>isCont f a; f a \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. sgn (f x)) a"
- unfolding isCont_def by (rule LIM_sgn)
+ unfolding isCont_def by (rule tendsto_sgn)
lemma isCont_setsum [simp]:
fixes f :: "'a \<Rightarrow> 'b::topological_space \<Rightarrow> 'c::real_normed_vector"
@@ -584,4 +490,29 @@
(X -- a --> (L::'b::topological_space))"
using LIMSEQ_SEQ_conv2 LIMSEQ_SEQ_conv1 ..
+subsection {* Legacy theorem names *}
+
+lemmas LIM_ident [simp] = tendsto_ident_at
+lemmas LIM_const [simp] = tendsto_const [where F="at x", standard]
+lemmas LIM_add = tendsto_add [where F="at x", standard]
+lemmas LIM_add_zero = tendsto_add_zero [where F="at x", standard]
+lemmas LIM_minus = tendsto_minus [where F="at x", standard]
+lemmas LIM_diff = tendsto_diff [where F="at x", standard]
+lemmas LIM_norm = tendsto_norm [where F="at x", standard]
+lemmas LIM_norm_zero = tendsto_norm_zero [where F="at x", standard]
+lemmas LIM_norm_zero_cancel = tendsto_norm_zero_cancel [where F="at x", standard]
+lemmas LIM_norm_zero_iff = tendsto_norm_zero_iff [where F="at x", standard]
+lemmas LIM_rabs = tendsto_rabs [where F="at x", standard]
+lemmas LIM_rabs_zero = tendsto_rabs_zero [where F="at x", standard]
+lemmas LIM_rabs_zero_cancel = tendsto_rabs_zero_cancel [where F="at x", standard]
+lemmas LIM_rabs_zero_iff = tendsto_rabs_zero_iff [where F="at x", standard]
+lemmas LIM_compose = tendsto_compose [where F="at x", standard]
+lemmas LIM_mult = tendsto_mult [where F="at x", standard]
+lemmas LIM_scaleR = tendsto_scaleR [where F="at x", standard]
+lemmas LIM_of_real = tendsto_of_real [where F="at x", standard]
+lemmas LIM_power = tendsto_power [where F="at x", standard]
+lemmas LIM_inverse = tendsto_inverse [where F="at x", standard]
+lemmas LIM_sgn = tendsto_sgn [where F="at x", standard]
+lemmas isCont_LIM_compose = isCont_tendsto_compose [where F="at x", standard]
+
end