header{* Limits and Continuity *}
theory Lim
imports SEQ
begin
text{*Standard Definitions*}
abbreviation
LIM :: "['a::topological_space => 'b::topological_space, 'a, 'b] => bool"
("((_)/ -- (_)/ --> (_))" [60, 0, 60] 60) where
"f -- a --> L ≡ (f ---> L) (at a)"
definition
isCont :: "['a::topological_space => 'b::topological_space, 'a] => bool" where
"isCont f a = (f -- a --> (f a))"
definition
isUCont :: "['a::metric_space => 'b::metric_space] => bool" where
"isUCont f = (∀r>0. ∃s>0. ∀x y. dist x y < s --> dist (f x) (f y) < r)"
subsection {* Limits of Functions *}
lemma LIM_def: "f -- a --> L =
(∀r > 0. ∃s > 0. ∀x. x ≠ a & dist x a < s
--> dist (f x) L < r)"
unfolding tendsto_iff eventually_at ..
lemma metric_LIM_I:
"(!!r. 0 < r ==> ∃s>0. ∀x. x ≠ a ∧ dist x a < s --> dist (f x) L < r)
==> f -- a --> L"
by (simp add: LIM_def)
lemma metric_LIM_D:
"[|f -- a --> L; 0 < r|]
==> ∃s>0. ∀x. x ≠ a ∧ dist x a < s --> dist (f x) L < r"
by (simp add: LIM_def)
lemma LIM_eq:
fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
shows "f -- a --> L =
(∀r>0.∃s>0.∀x. x ≠ a & norm (x-a) < s --> norm (f x - L) < r)"
by (simp add: LIM_def dist_norm)
lemma LIM_I:
fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
shows "(!!r. 0<r ==> ∃s>0.∀x. x ≠ a & norm (x-a) < s --> norm (f x - L) < r)
==> f -- a --> L"
by (simp add: LIM_eq)
lemma LIM_D:
fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
shows "[| f -- a --> L; 0<r |]
==> ∃s>0.∀x. x ≠ a & norm (x-a) < s --> norm (f x - L) < r"
by (simp add: LIM_eq)
lemma LIM_offset:
fixes a :: "'a::real_normed_vector"
shows "f -- a --> L ==> (λx. f (x + k)) -- a - k --> L"
apply (rule topological_tendstoI)
apply (drule (2) topological_tendstoD)
apply (simp only: eventually_at dist_norm)
apply (clarify, rule_tac x=d in exI, safe)
apply (drule_tac x="x + k" in spec)
apply (simp add: algebra_simps)
done
lemma LIM_offset_zero:
fixes a :: "'a::real_normed_vector"
shows "f -- a --> L ==> (λh. f (a + h)) -- 0 --> L"
by (drule_tac k="a" in LIM_offset, simp add: add_commute)
lemma LIM_offset_zero_cancel:
fixes a :: "'a::real_normed_vector"
shows "(λh. f (a + h)) -- 0 --> L ==> f -- a --> L"
by (drule_tac k="- a" in LIM_offset, simp)
lemma LIM_cong_limit: "[| f -- x --> L ; K = L |] ==> f -- x --> K" by simp
lemma LIM_zero:
fixes f :: "'a::topological_space => 'b::real_normed_vector"
shows "(f ---> l) F ==> ((λx. f x - l) ---> 0) F"
unfolding tendsto_iff dist_norm by simp
lemma LIM_zero_cancel:
fixes f :: "'a::topological_space => 'b::real_normed_vector"
shows "((λx. f x - l) ---> 0) F ==> (f ---> l) F"
unfolding tendsto_iff dist_norm by simp
lemma LIM_zero_iff:
fixes f :: "'a::metric_space => 'b::real_normed_vector"
shows "((λx. f x - l) ---> 0) F = (f ---> l) F"
unfolding tendsto_iff dist_norm by simp
lemma metric_LIM_imp_LIM:
assumes f: "f -- a --> l"
assumes le: "!!x. x ≠ a ==> dist (g x) m ≤ dist (f x) l"
shows "g -- a --> m"
by (rule metric_tendsto_imp_tendsto [OF f],
auto simp add: eventually_at_topological le)
lemma LIM_imp_LIM:
fixes f :: "'a::topological_space => 'b::real_normed_vector"
fixes g :: "'a::topological_space => 'c::real_normed_vector"
assumes f: "f -- a --> l"
assumes le: "!!x. x ≠ a ==> norm (g x - m) ≤ norm (f x - l)"
shows "g -- a --> m"
by (rule metric_LIM_imp_LIM [OF f],
simp add: dist_norm le)
lemma LIM_const_not_eq:
fixes a :: "'a::perfect_space"
fixes k L :: "'b::t2_space"
shows "k ≠ L ==> ¬ (λx. k) -- a --> L"
by (simp add: tendsto_const_iff)
lemmas LIM_not_zero = LIM_const_not_eq [where L = 0]
lemma LIM_const_eq:
fixes a :: "'a::perfect_space"
fixes k L :: "'b::t2_space"
shows "(λx. k) -- a --> L ==> k = L"
by (simp add: tendsto_const_iff)
lemma LIM_unique:
fixes a :: "'a::perfect_space"
fixes L M :: "'b::t2_space"
shows "[|f -- a --> L; f -- a --> M|] ==> L = M"
using at_neq_bot by (rule tendsto_unique)
text{*Limits are equal for functions equal except at limit point*}
lemma LIM_equal:
"[| ∀x. x ≠ a --> (f x = g x) |] ==> (f -- a --> l) = (g -- a --> l)"
unfolding tendsto_def eventually_at_topological by simp
lemma LIM_cong:
"[|a = b; !!x. x ≠ b ==> f x = g x; l = m|]
==> ((λx. f x) -- a --> l) = ((λx. g x) -- b --> m)"
by (simp add: LIM_equal)
lemma metric_LIM_equal2:
assumes 1: "0 < R"
assumes 2: "!!x. [|x ≠ a; dist x a < R|] ==> f x = g x"
shows "g -- a --> l ==> f -- a --> l"
apply (rule topological_tendstoI)
apply (drule (2) topological_tendstoD)
apply (simp add: eventually_at, safe)
apply (rule_tac x="min d R" in exI, safe)
apply (simp add: 1)
apply (simp add: 2)
done
lemma LIM_equal2:
fixes f g :: "'a::real_normed_vector => 'b::topological_space"
assumes 1: "0 < R"
assumes 2: "!!x. [|x ≠ a; norm (x - a) < R|] ==> f x = g x"
shows "g -- a --> l ==> f -- a --> l"
by (rule metric_LIM_equal2 [OF 1 2], simp_all add: dist_norm)
lemma LIM_compose_eventually:
assumes f: "f -- a --> b"
assumes g: "g -- b --> c"
assumes inj: "eventually (λx. f x ≠ b) (at a)"
shows "(λx. g (f x)) -- a --> c"
using g f inj by (rule tendsto_compose_eventually)
lemma metric_LIM_compose2:
assumes f: "f -- a --> b"
assumes g: "g -- b --> c"
assumes inj: "∃d>0. ∀x. x ≠ a ∧ dist x a < d --> f x ≠ b"
shows "(λx. g (f x)) -- a --> c"
using g f inj [folded eventually_at]
by (rule tendsto_compose_eventually)
lemma LIM_compose2:
fixes a :: "'a::real_normed_vector"
assumes f: "f -- a --> b"
assumes g: "g -- b --> c"
assumes inj: "∃d>0. ∀x. x ≠ a ∧ norm (x - a) < d --> f x ≠ b"
shows "(λx. g (f x)) -- a --> c"
by (rule metric_LIM_compose2 [OF f g inj [folded dist_norm]])
lemma LIM_o: "[|g -- l --> g l; f -- a --> l|] ==> (g o f) -- a --> g l"
unfolding o_def by (rule tendsto_compose)
lemma real_LIM_sandwich_zero:
fixes f g :: "'a::topological_space => real"
assumes f: "f -- a --> 0"
assumes 1: "!!x. x ≠ a ==> 0 ≤ g x"
assumes 2: "!!x. x ≠ a ==> g x ≤ f x"
shows "g -- a --> 0"
proof (rule LIM_imp_LIM [OF f])
fix x assume x: "x ≠ a"
have "norm (g x - 0) = g x" by (simp add: 1 x)
also have "g x ≤ f x" by (rule 2 [OF x])
also have "f x ≤ ¦f x¦" by (rule abs_ge_self)
also have "¦f x¦ = norm (f x - 0)" by simp
finally show "norm (g x - 0) ≤ norm (f x - 0)" .
qed
subsection {* Continuity *}
lemma LIM_isCont_iff:
fixes f :: "'a::real_normed_vector => 'b::topological_space"
shows "(f -- a --> f a) = ((λh. f (a + h)) -- 0 --> f a)"
by (rule iffI [OF LIM_offset_zero LIM_offset_zero_cancel])
lemma isCont_iff:
fixes f :: "'a::real_normed_vector => 'b::topological_space"
shows "isCont f x = (λh. f (x + h)) -- 0 --> f x"
by (simp add: isCont_def LIM_isCont_iff)
lemma isCont_ident [simp]: "isCont (λx. x) a"
unfolding isCont_def by (rule tendsto_ident_at)
lemma isCont_const [simp]: "isCont (λx. k) a"
unfolding isCont_def by (rule tendsto_const)
lemma isCont_norm [simp]:
fixes f :: "'a::topological_space => 'b::real_normed_vector"
shows "isCont f a ==> isCont (λx. norm (f x)) a"
unfolding isCont_def by (rule tendsto_norm)
lemma isCont_rabs [simp]:
fixes f :: "'a::topological_space => real"
shows "isCont f a ==> isCont (λx. ¦f x¦) a"
unfolding isCont_def by (rule tendsto_rabs)
lemma isCont_add [simp]:
fixes f :: "'a::topological_space => 'b::real_normed_vector"
shows "[|isCont f a; isCont g a|] ==> isCont (λx. f x + g x) a"
unfolding isCont_def by (rule tendsto_add)
lemma isCont_minus [simp]:
fixes f :: "'a::topological_space => 'b::real_normed_vector"
shows "isCont f a ==> isCont (λx. - f x) a"
unfolding isCont_def by (rule tendsto_minus)
lemma isCont_diff [simp]:
fixes f :: "'a::topological_space => 'b::real_normed_vector"
shows "[|isCont f a; isCont g a|] ==> isCont (λx. f x - g x) a"
unfolding isCont_def by (rule tendsto_diff)
lemma isCont_mult [simp]:
fixes f g :: "'a::topological_space => 'b::real_normed_algebra"
shows "[|isCont f a; isCont g a|] ==> isCont (λx. f x * g x) a"
unfolding isCont_def by (rule tendsto_mult)
lemma isCont_inverse [simp]:
fixes f :: "'a::topological_space => 'b::real_normed_div_algebra"
shows "[|isCont f a; f a ≠ 0|] ==> isCont (λx. inverse (f x)) a"
unfolding isCont_def by (rule tendsto_inverse)
lemma isCont_divide [simp]:
fixes f g :: "'a::topological_space => 'b::real_normed_field"
shows "[|isCont f a; isCont g a; g a ≠ 0|] ==> isCont (λx. f x / g x) a"
unfolding isCont_def by (rule tendsto_divide)
lemma isCont_tendsto_compose:
"[|isCont g l; (f ---> l) F|] ==> ((λx. g (f x)) ---> g l) F"
unfolding isCont_def by (rule tendsto_compose)
lemma metric_isCont_LIM_compose2:
assumes f [unfolded isCont_def]: "isCont f a"
assumes g: "g -- f a --> l"
assumes inj: "∃d>0. ∀x. x ≠ a ∧ dist x a < d --> f x ≠ f a"
shows "(λx. g (f x)) -- a --> l"
by (rule metric_LIM_compose2 [OF f g inj])
lemma isCont_LIM_compose2:
fixes a :: "'a::real_normed_vector"
assumes f [unfolded isCont_def]: "isCont f a"
assumes g: "g -- f a --> l"
assumes inj: "∃d>0. ∀x. x ≠ a ∧ norm (x - a) < d --> f x ≠ f a"
shows "(λx. g (f x)) -- a --> l"
by (rule LIM_compose2 [OF f g inj])
lemma isCont_o2: "[|isCont f a; isCont g (f a)|] ==> isCont (λx. g (f x)) a"
unfolding isCont_def by (rule tendsto_compose)
lemma isCont_o: "[|isCont f a; isCont g (f a)|] ==> isCont (g o f) a"
unfolding o_def by (rule isCont_o2)
lemma (in bounded_linear) isCont:
"isCont g a ==> isCont (λx. f (g x)) a"
unfolding isCont_def by (rule tendsto)
lemma (in bounded_bilinear) isCont:
"[|isCont f a; isCont g a|] ==> isCont (λx. f x ** g x) a"
unfolding isCont_def by (rule tendsto)
lemmas isCont_scaleR [simp] =
bounded_bilinear.isCont [OF bounded_bilinear_scaleR]
lemmas isCont_of_real [simp] =
bounded_linear.isCont [OF bounded_linear_of_real]
lemma isCont_power [simp]:
fixes f :: "'a::topological_space => 'b::{power,real_normed_algebra}"
shows "isCont f a ==> isCont (λx. f x ^ n) a"
unfolding isCont_def by (rule tendsto_power)
lemma isCont_sgn [simp]:
fixes f :: "'a::topological_space => 'b::real_normed_vector"
shows "[|isCont f a; f a ≠ 0|] ==> isCont (λx. sgn (f x)) a"
unfolding isCont_def by (rule tendsto_sgn)
lemma isCont_setsum [simp]:
fixes f :: "'a => 'b::topological_space => 'c::real_normed_vector"
fixes A :: "'a set"
shows "∀i∈A. isCont (f i) a ==> isCont (λx. ∑i∈A. f i x) a"
unfolding isCont_def by (simp add: tendsto_setsum)
lemmas isCont_intros =
isCont_ident isCont_const isCont_norm isCont_rabs isCont_add isCont_minus
isCont_diff isCont_mult isCont_inverse isCont_divide isCont_scaleR
isCont_of_real isCont_power isCont_sgn isCont_setsum
lemma LIM_less_bound: fixes f :: "real => real" assumes "b < x"
and all_le: "∀ x' ∈ { b <..< x}. 0 ≤ f x'" and isCont: "isCont f x"
shows "0 ≤ f x"
proof (rule ccontr)
assume "¬ 0 ≤ f x" hence "f x < 0" by auto
hence "0 < - f x / 2" by auto
from isCont[unfolded isCont_def, THEN LIM_D, OF this]
obtain s where "s > 0" and s_D: "!!x'. [| x' ≠ x ; ¦ x' - x ¦ < s |] ==> ¦ f x' - f x ¦ < - f x / 2" by auto
let ?x = "x - min (s / 2) ((x - b) / 2)"
have "?x < x" and "¦ ?x - x ¦ < s"
using `b < x` and `0 < s` by auto
have "b < ?x"
proof (cases "s < x - b")
case True thus ?thesis using `0 < s` by auto
next
case False hence "s / 2 ≥ (x - b) / 2" by auto
hence "?x = (x + b) / 2" by (simp add: field_simps min_max.inf_absorb2)
thus ?thesis using `b < x` by auto
qed
hence "0 ≤ f ?x" using all_le `?x < x` by auto
moreover have "¦f ?x - f x¦ < - f x / 2"
using s_D[OF _ `¦ ?x - x ¦ < s`] `?x < x` by auto
hence "f ?x - f x < - f x / 2" by auto
hence "f ?x < f x / 2" by auto
hence "f ?x < 0" using `f x < 0` by auto
thus False using `0 ≤ f ?x` by auto
qed
subsection {* Uniform Continuity *}
lemma isUCont_isCont: "isUCont f ==> isCont f x"
by (simp add: isUCont_def isCont_def LIM_def, force)
lemma isUCont_Cauchy:
"[|isUCont f; Cauchy X|] ==> Cauchy (λn. f (X n))"
unfolding isUCont_def
apply (rule metric_CauchyI)
apply (drule_tac x=e in spec, safe)
apply (drule_tac e=s in metric_CauchyD, safe)
apply (rule_tac x=M in exI, simp)
done
lemma (in bounded_linear) isUCont: "isUCont f"
unfolding isUCont_def dist_norm
proof (intro allI impI)
fix r::real assume r: "0 < r"
obtain K where K: "0 < K" and norm_le: "!!x. norm (f x) ≤ norm x * K"
using pos_bounded by fast
show "∃s>0. ∀x y. norm (x - y) < s --> norm (f x - f y) < r"
proof (rule exI, safe)
from r K show "0 < r / K" by (rule divide_pos_pos)
next
fix x y :: 'a
assume xy: "norm (x - y) < r / K"
have "norm (f x - f y) = norm (f (x - y))" by (simp only: diff)
also have "… ≤ norm (x - y) * K" by (rule norm_le)
also from K xy have "… < r" by (simp only: pos_less_divide_eq)
finally show "norm (f x - f y) < r" .
qed
qed
lemma (in bounded_linear) Cauchy: "Cauchy X ==> Cauchy (λn. f (X n))"
by (rule isUCont [THEN isUCont_Cauchy])
subsection {* Relation of LIM and LIMSEQ *}
lemma sequentially_imp_eventually_within:
fixes a :: "'a::metric_space"
assumes "∀f. (∀n. f n ∈ s ∧ f n ≠ a) ∧ f ----> a -->
eventually (λn. P (f n)) sequentially"
shows "eventually P (at a within s)"
proof (rule ccontr)
let ?I = "λn. inverse (real (Suc n))"
def F ≡ "λn::nat. SOME x. x ∈ s ∧ x ≠ a ∧ dist x a < ?I n ∧ ¬ P x"
assume "¬ eventually P (at a within s)"
hence P: "∀d>0. ∃x. x ∈ s ∧ x ≠ a ∧ dist x a < d ∧ ¬ P x"
unfolding Limits.eventually_within Limits.eventually_at by fast
hence "!!n. ∃x. x ∈ s ∧ x ≠ a ∧ dist x a < ?I n ∧ ¬ P x" by simp
hence F: "!!n. F n ∈ s ∧ F n ≠ a ∧ dist (F n) a < ?I n ∧ ¬ P (F n)"
unfolding F_def by (rule someI_ex)
hence F0: "∀n. F n ∈ s" and F1: "∀n. F n ≠ a"
and F2: "∀n. dist (F n) a < ?I n" and F3: "∀n. ¬ P (F n)"
by fast+
from LIMSEQ_inverse_real_of_nat have "F ----> a"
by (rule metric_tendsto_imp_tendsto,
simp add: dist_norm F2 less_imp_le)
hence "eventually (λn. P (F n)) sequentially"
using assms F0 F1 by simp
thus "False" by (simp add: F3)
qed
lemma sequentially_imp_eventually_at:
fixes a :: "'a::metric_space"
assumes "∀f. (∀n. f n ≠ a) ∧ f ----> a -->
eventually (λn. P (f n)) sequentially"
shows "eventually P (at a)"
using assms sequentially_imp_eventually_within [where s=UNIV] by simp
lemma LIMSEQ_SEQ_conv1:
fixes f :: "'a::topological_space => 'b::topological_space"
assumes f: "f -- a --> l"
shows "∀S. (∀n. S n ≠ a) ∧ S ----> a --> (λn. f (S n)) ----> l"
using tendsto_compose_eventually [OF f, where F=sequentially] by simp
lemma LIMSEQ_SEQ_conv2:
fixes f :: "'a::metric_space => 'b::topological_space"
assumes "∀S. (∀n. S n ≠ a) ∧ S ----> a --> (λn. f (S n)) ----> l"
shows "f -- a --> l"
using assms unfolding tendsto_def [where l=l]
by (simp add: sequentially_imp_eventually_at)
lemma LIMSEQ_SEQ_conv:
"(∀S. (∀n. S n ≠ a) ∧ S ----> (a::'a::metric_space) --> (λn. X (S n)) ----> L) =
(X -- a --> (L::'b::topological_space))"
using LIMSEQ_SEQ_conv2 LIMSEQ_SEQ_conv1 ..
end