# Theory HLim

theory HLim
imports Star
```(*  Title:      HOL/NSA/HLim.thy
Author:     Jacques D. Fleuriot, University of Cambridge
Author:     Lawrence C Paulson
*)

section‹Limits and Continuity (Nonstandard)›

theory HLim
imports Star
begin

text‹Nonstandard Definitions›

definition
NSLIM :: "['a::real_normed_vector => 'b::real_normed_vector, 'a, 'b] => bool"
("((_)/ ─(_)/→⇩N⇩S (_))" [60, 0, 60] 60) where
"f ─a→⇩N⇩S L =
(∀x. (x ≠ star_of a & x ≈ star_of a --> ( *f* f) x ≈ star_of L))"

definition
isNSCont :: "['a::real_normed_vector => 'b::real_normed_vector, 'a] => bool" where
―‹NS definition dispenses with limit notions›
"isNSCont f a = (∀y. y ≈ star_of a -->
( *f* f) y ≈ star_of (f a))"

definition
isNSUCont :: "['a::real_normed_vector => 'b::real_normed_vector] => bool" where
"isNSUCont f = (∀x y. x ≈ y --> ( *f* f) x ≈ ( *f* f) y)"

subsection ‹Limits of Functions›

lemma NSLIM_I:
"(⋀x. ⟦x ≠ star_of a; x ≈ star_of a⟧ ⟹ starfun f x ≈ star_of L)
⟹ f ─a→⇩N⇩S L"

lemma NSLIM_D:
"⟦f ─a→⇩N⇩S L; x ≠ star_of a; x ≈ star_of a⟧
⟹ starfun f x ≈ star_of L"

text‹Proving properties of limits using nonstandard definition.
The properties hold for standard limits as well!›

lemma NSLIM_mult:
fixes l m :: "'a::real_normed_algebra"
shows "[| f ─x→⇩N⇩S l; g ─x→⇩N⇩S m |]
==> (%x. f(x) * g(x)) ─x→⇩N⇩S (l * m)"
by (auto simp add: NSLIM_def intro!: approx_mult_HFinite)

lemma starfun_scaleR [simp]:
"starfun (λx. f x *⇩R g x) = (λx. scaleHR (starfun f x) (starfun g x))"
by transfer (rule refl)

lemma NSLIM_scaleR:
"[| f ─x→⇩N⇩S l; g ─x→⇩N⇩S m |]
==> (%x. f(x) *⇩R g(x)) ─x→⇩N⇩S (l *⇩R m)"
by (auto simp add: NSLIM_def intro!: approx_scaleR_HFinite)

"[| f ─x→⇩N⇩S l; g ─x→⇩N⇩S m |]
==> (%x. f(x) + g(x)) ─x→⇩N⇩S (l + m)"

lemma NSLIM_const [simp]: "(%x. k) ─x→⇩N⇩S k"

lemma NSLIM_minus: "f ─a→⇩N⇩S L ==> (%x. -f(x)) ─a→⇩N⇩S -L"

lemma NSLIM_diff:
"⟦f ─x→⇩N⇩S l; g ─x→⇩N⇩S m⟧ ⟹ (λx. f x - g x) ─x→⇩N⇩S (l - m)"

lemma NSLIM_add_minus: "[| f ─x→⇩N⇩S l; g ─x→⇩N⇩S m |] ==> (%x. f(x) + -g(x)) ─x→⇩N⇩S (l + -m)"

lemma NSLIM_inverse:
fixes L :: "'a::real_normed_div_algebra"
shows "[| f ─a→⇩N⇩S L;  L ≠ 0 |]
==> (%x. inverse(f(x))) ─a→⇩N⇩S (inverse L)"
apply (drule spec)
done

lemma NSLIM_zero:
assumes f: "f ─a→⇩N⇩S l" shows "(%x. f(x) - l) ─a→⇩N⇩S 0"
proof -
have "(λx. f x - l) ─a→⇩N⇩S l - l"
by (rule NSLIM_diff [OF f NSLIM_const])
thus ?thesis by simp
qed

lemma NSLIM_zero_cancel: "(%x. f(x) - l) ─x→⇩N⇩S 0 ==> f ─x→⇩N⇩S l"
apply (drule_tac g = "%x. l" and m = l in NSLIM_add)
done

lemma NSLIM_const_not_eq:
fixes a :: "'a::real_normed_algebra_1"
shows "k ≠ L ⟹ ¬ (λx. k) ─a→⇩N⇩S L"
apply (rule_tac x="star_of a + of_hypreal ε" in exI)
done

lemma NSLIM_not_zero:
fixes a :: "'a::real_normed_algebra_1"
shows "k ≠ 0 ⟹ ¬ (λx. k) ─a→⇩N⇩S 0"
by (rule NSLIM_const_not_eq)

lemma NSLIM_const_eq:
fixes a :: "'a::real_normed_algebra_1"
shows "(λx. k) ─a→⇩N⇩S L ⟹ k = L"
apply (rule ccontr)
apply (blast dest: NSLIM_const_not_eq)
done

lemma NSLIM_unique:
fixes a :: "'a::real_normed_algebra_1"
shows "⟦f ─a→⇩N⇩S L; f ─a→⇩N⇩S M⟧ ⟹ L = M"
apply (drule (1) NSLIM_diff)
apply (auto dest!: NSLIM_const_eq)
done

lemma NSLIM_mult_zero:
fixes f g :: "'a::real_normed_vector ⇒ 'b::real_normed_algebra"
shows "[| f ─x→⇩N⇩S 0; g ─x→⇩N⇩S 0 |] ==> (%x. f(x)*g(x)) ─x→⇩N⇩S 0"
by (drule NSLIM_mult, auto)

lemma NSLIM_self: "(%x. x) ─a→⇩N⇩S a"

subsubsection ‹Equivalence of @{term filterlim} and @{term NSLIM}›

lemma LIM_NSLIM:
assumes f: "f ─a→ L" shows "f ─a→⇩N⇩S L"
proof (rule NSLIM_I)
fix x
assume neq: "x ≠ star_of a"
assume approx: "x ≈ star_of a"
have "starfun f x - star_of L ∈ Infinitesimal"
proof (rule InfinitesimalI2)
fix r::real assume r: "0 < r"
from LIM_D [OF f r]
obtain s where s: "0 < s" and
less_r: "⋀x. ⟦x ≠ a; norm (x - a) < s⟧ ⟹ norm (f x - L) < r"
by fast
from less_r have less_r':
"⋀x. ⟦x ≠ star_of a; hnorm (x - star_of a) < star_of s⟧
⟹ hnorm (starfun f x - star_of L) < star_of r"
by transfer
from approx have "x - star_of a ∈ Infinitesimal"
by (unfold approx_def)
hence "hnorm (x - star_of a) < star_of s"
using s by (rule InfinitesimalD2)
with neq show "hnorm (starfun f x - star_of L) < star_of r"
by (rule less_r')
qed
thus "starfun f x ≈ star_of L"
by (unfold approx_def)
qed

lemma NSLIM_LIM:
assumes f: "f ─a→⇩N⇩S L" shows "f ─a→ L"
proof (rule LIM_I)
fix r::real assume r: "0 < r"
have "∃s>0. ∀x. x ≠ star_of a ∧ hnorm (x - star_of a) < s
⟶ hnorm (starfun f x - star_of L) < star_of r"
proof (rule exI, safe)
show "0 < ε" by (rule hypreal_epsilon_gt_zero)
next
fix x assume neq: "x ≠ star_of a"
assume "hnorm (x - star_of a) < ε"
with Infinitesimal_epsilon
have "x - star_of a ∈ Infinitesimal"
by (rule hnorm_less_Infinitesimal)
hence "x ≈ star_of a"
by (unfold approx_def)
with f neq have "starfun f x ≈ star_of L"
by (rule NSLIM_D)
hence "starfun f x - star_of L ∈ Infinitesimal"
by (unfold approx_def)
thus "hnorm (starfun f x - star_of L) < star_of r"
using r by (rule InfinitesimalD2)
qed
thus "∃s>0. ∀x. x ≠ a ∧ norm (x - a) < s ⟶ norm (f x - L) < r"
by transfer
qed

theorem LIM_NSLIM_iff: "(f ─x→ L) = (f ─x→⇩N⇩S L)"
by (blast intro: LIM_NSLIM NSLIM_LIM)

subsection ‹Continuity›

lemma isNSContD:
"⟦isNSCont f a; y ≈ star_of a⟧ ⟹ ( *f* f) y ≈ star_of (f a)"

lemma isNSCont_NSLIM: "isNSCont f a ==> f ─a→⇩N⇩S (f a) "

lemma NSLIM_isNSCont: "f ─a→⇩N⇩S (f a) ==> isNSCont f a"
apply (simp add: isNSCont_def NSLIM_def, auto)
apply (case_tac "y = star_of a", auto)
done

text‹NS continuity can be defined using NS Limit in
similar fashion to standard def of continuity›
lemma isNSCont_NSLIM_iff: "(isNSCont f a) = (f ─a→⇩N⇩S (f a))"
by (blast intro: isNSCont_NSLIM NSLIM_isNSCont)

text‹Hence, NS continuity can be given
in terms of standard limit›
lemma isNSCont_LIM_iff: "(isNSCont f a) = (f ─a→ (f a))"

text‹Moreover, it's trivial now that NS continuity
is equivalent to standard continuity›
lemma isNSCont_isCont_iff: "(isNSCont f a) = (isCont f a)"
apply (rule isNSCont_LIM_iff)
done

text‹Standard continuity ==> NS continuity›
lemma isCont_isNSCont: "isCont f a ==> isNSCont f a"
by (erule isNSCont_isCont_iff [THEN iffD2])

text‹NS continuity ==> Standard continuity›
lemma isNSCont_isCont: "isNSCont f a ==> isCont f a"
by (erule isNSCont_isCont_iff [THEN iffD1])

text‹Alternative definition of continuity›

(* Prove equivalence between NS limits - *)
(* seems easier than using standard def  *)
lemma NSLIM_h_iff: "(f ─a→⇩N⇩S L) = ((%h. f(a + h)) ─0→⇩N⇩S L)"
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])
apply (rule_tac x = x in star_cases)
apply (rule_tac [2] x = x in star_cases)
done

lemma NSLIM_isCont_iff: "(f ─a→⇩N⇩S f a) = ((%h. f(a + h)) ─0→⇩N⇩S f a)"
by (fact NSLIM_h_iff)

lemma isNSCont_minus: "isNSCont f a ==> isNSCont (%x. - f x) a"

lemma isNSCont_inverse:
fixes f :: "'a::real_normed_vector ⇒ 'b::real_normed_div_algebra"
shows "[| isNSCont f x; f x ≠ 0 |] ==> isNSCont (%x. inverse (f x)) x"
by (auto intro: isCont_inverse simp add: isNSCont_isCont_iff)

lemma isNSCont_const [simp]: "isNSCont (%x. k) a"

lemma isNSCont_abs [simp]: "isNSCont abs (a::real)"
apply (auto intro: approx_hrabs simp add: starfun_rabs_hrabs)
done

subsection ‹Uniform Continuity›

lemma isNSUContD: "[| isNSUCont f; x ≈ y|] ==> ( *f* f) x ≈ ( *f* f) y"

lemma isUCont_isNSUCont:
fixes f :: "'a::real_normed_vector ⇒ 'b::real_normed_vector"
assumes f: "isUCont f" shows "isNSUCont f"
proof (unfold isNSUCont_def, safe)
fix x y :: "'a star"
assume approx: "x ≈ y"
have "starfun f x - starfun f y ∈ Infinitesimal"
proof (rule InfinitesimalI2)
fix r::real assume r: "0 < r"
with f obtain s where s: "0 < s" and
less_r: "⋀x y. norm (x - y) < s ⟹ norm (f x - f y) < r"
by (auto simp add: isUCont_def dist_norm)
from less_r have less_r':
"⋀x y. hnorm (x - y) < star_of s
⟹ hnorm (starfun f x - starfun f y) < star_of r"
by transfer
from approx have "x - y ∈ Infinitesimal"
by (unfold approx_def)
hence "hnorm (x - y) < star_of s"
using s by (rule InfinitesimalD2)
thus "hnorm (starfun f x - starfun f y) < star_of r"
by (rule less_r')
qed
thus "starfun f x ≈ starfun f y"
by (unfold approx_def)
qed

lemma isNSUCont_isUCont:
fixes f :: "'a::real_normed_vector ⇒ 'b::real_normed_vector"
assumes f: "isNSUCont f" shows "isUCont f"
proof (unfold isUCont_def dist_norm, safe)
fix r::real assume r: "0 < r"
have "∃s>0. ∀x y. hnorm (x - y) < s
⟶ hnorm (starfun f x - starfun f y) < star_of r"
proof (rule exI, safe)
show "0 < ε" by (rule hypreal_epsilon_gt_zero)
next
fix x y :: "'a star"
assume "hnorm (x - y) < ε"
with Infinitesimal_epsilon
have "x - y ∈ Infinitesimal"
by (rule hnorm_less_Infinitesimal)
hence "x ≈ y"
by (unfold approx_def)
with f have "starfun f x ≈ starfun f y"