# Theory HDeriv

```(*  Title:      HOL/Nonstandard_Analysis/HDeriv.thy
Author:     Jacques D. Fleuriot
Conversion to Isar and new proofs by Lawrence C Paulson, 2004
*)

section ‹Differentiation (Nonstandard)›

theory HDeriv
imports HLim
begin

text ‹Nonstandard Definitions.›

definition nsderiv :: "['a::real_normed_field ⇒ 'a, 'a, 'a] ⇒ bool"
("(NSDERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60)
where "NSDERIV f x :> D ⟷
(∀h ∈ Infinitesimal - {0}. (( *f* f)(star_of x + h) - star_of (f x)) / h ≈ star_of D)"

definition NSdifferentiable :: "['a::real_normed_field ⇒ 'a, 'a] ⇒ bool"
(infixl "NSdifferentiable" 60)
where "f NSdifferentiable x ⟷ (∃D. NSDERIV f x :> D)"

definition increment :: "(real ⇒ real) ⇒ real ⇒ hypreal ⇒ hypreal"
where "increment f x h =
(SOME inc. f NSdifferentiable x ∧ inc = ( *f* f) (hypreal_of_real x + h) - hypreal_of_real (f x))"

subsection ‹Derivatives›

lemma DERIV_NS_iff: "(DERIV f x :> D) ⟷ (λh. (f (x + h) - f x) / h) ─0→⇩N⇩S D"

lemma NS_DERIV_D: "DERIV f x :> D ⟹ (λh. (f (x + h) - f x) / h) ─0→⇩N⇩S D"

lemma Infinitesimal_of_hypreal:
"x ∈ Infinitesimal ⟹ (( *f* of_real) x::'a::real_normed_div_algebra star) ∈ Infinitesimal"
by (metis Infinitesimal_of_hypreal_iff of_hypreal_def)

lemma of_hypreal_eq_0_iff: "⋀x. (( *f* of_real) x = (0::'a::real_algebra_1 star)) = (x = 0)"
by transfer (rule of_real_eq_0_iff)

lemma NSDeriv_unique:
assumes "NSDERIV f x :> D" "NSDERIV f x :> E"
shows "NSDERIV f x :> D ⟹ NSDERIV f x :> E ⟹ D = E"
proof -
have "∃s. (s::'a star) ∈ Infinitesimal - {0}"
by (metis Diff_iff HDeriv.of_hypreal_eq_0_iff Infinitesimal_epsilon Infinitesimal_of_hypreal epsilon_not_zero singletonD)
with assms show ?thesis
by (meson approx_trans3 nsderiv_def star_of_approx_iff)
qed

text ‹First ‹NSDERIV› in terms of ‹NSLIM›.›

text ‹First equivalence.›
lemma NSDERIV_NSLIM_iff: "(NSDERIV f x :> D) ⟷ (λh. (f (x + h) - f x) / h) ─0→⇩N⇩S D"
by (auto simp add: nsderiv_def NSLIM_def starfun_lambda_cancel mem_infmal_iff)

text ‹Second equivalence.›
lemma NSDERIV_NSLIM_iff2: "(NSDERIV f x :> D) ⟷ (λz. (f z - f x) / (z - x)) ─x→⇩N⇩S D"
by (simp add: NSDERIV_NSLIM_iff DERIV_LIM_iff LIM_NSLIM_iff [symmetric])

text ‹While we're at it!›
lemma NSDERIV_iff2:
"(NSDERIV f x :> D) ⟷
(∀w. w ≠ star_of x ∧ w ≈ star_of x ⟶ ( *f* (λz. (f z - f x) / (z - x))) w ≈ star_of D)"

lemma NSDERIVD5:
"⟦NSDERIV f x :> D; u ≈ hypreal_of_real x⟧ ⟹
( *f* (λz. f z - f x)) u ≈ hypreal_of_real D * (u - hypreal_of_real x)"
unfolding NSDERIV_iff2
apply (case_tac "u = hypreal_of_real x", auto)
by (metis (mono_tags, lifting) HFinite_star_of Infinitesimal_ratio approx_def approx_minus_iff approx_mult_subst approx_star_of_HFinite approx_sym mult_zero_right right_minus_eq)

lemma NSDERIVD4:
"⟦NSDERIV f x :> D; h ∈ Infinitesimal⟧
⟹ ( *f* f)(hypreal_of_real x + h) - hypreal_of_real (f x) ≈ hypreal_of_real D * h"
apply (case_tac "h = 0", simp)
by (meson DiffI Infinitesimal_approx Infinitesimal_ratio Infinitesimal_star_of_mult2 approx_star_of_HFinite singletonD)

text ‹Differentiability implies continuity nice and simple "algebraic" proof.›
lemma NSDERIV_isNSCont:
assumes "NSDERIV f x :> D" shows "isNSCont f x"
unfolding isNSCont_NSLIM_iff NSLIM_def
proof clarify
fix x'
assume "x' ≠ star_of x" "x' ≈ star_of x"
then have m0: "x' - star_of x ∈ Infinitesimal - {0}"
using bex_Infinitesimal_iff by auto
then have "(( *f* f) x' - star_of (f x)) / (x' - star_of x) ≈ star_of D"
by (metis ‹x' ≈ star_of x› add_diff_cancel_left' assms bex_Infinitesimal_iff2 nsderiv_def)
then have "(( *f* f) x' - star_of (f x)) / (x' - star_of x) ∈ HFinite"
by (metis approx_star_of_HFinite)
then show "( *f* f) x' ≈ star_of (f x)"
by (metis (no_types) Diff_iff Infinitesimal_ratio m0 bex_Infinitesimal_iff insert_iff)
qed

text ‹Differentiation rules for combinations of functions
follow from clear, straightforward, algebraic manipulations.›

text ‹Constant function.›

(* use simple constant nslimit theorem *)
lemma NSDERIV_const [simp]: "NSDERIV (λx. k) x :> 0"

text ‹Sum of functions- proved easily.›

assumes "NSDERIV f x :> Da" "NSDERIV g x :> Db"
shows "NSDERIV (λx. f x + g x) x :> Da + Db"
proof -
have "((λx. f x + g x) has_field_derivative Da + Db) (at x)"
using assms DERIV_NS_iff NSDERIV_NSLIM_iff field_differentiable_add by blast
then show ?thesis
qed

text ‹Product of functions - Proof is simple.›

lemma NSDERIV_mult:
assumes "NSDERIV g x :> Db" "NSDERIV f x :> Da"
shows "NSDERIV (λx. f x * g x) x :> (Da * g x) + (Db * f x)"
proof -
have "(f has_field_derivative Da) (at x)" "(g has_field_derivative Db) (at x)"
using assms by (simp_all add: DERIV_NS_iff NSDERIV_NSLIM_iff)
then have "((λa. f a * g a) has_field_derivative Da * g x + Db * f x) (at x)"
using DERIV_mult by blast
then show ?thesis
qed

text ‹Multiplying by a constant.›
lemma NSDERIV_cmult: "NSDERIV f x :> D ⟹ NSDERIV (λx. c * f x) x :> c * D"
unfolding times_divide_eq_right [symmetric] NSDERIV_NSLIM_iff
minus_mult_right right_diff_distrib [symmetric]
by (erule NSLIM_const [THEN NSLIM_mult])

text ‹Negation of function.›
lemma NSDERIV_minus: "NSDERIV f x :> D ⟹ NSDERIV (λx. - f x) x :> - D"
assume "(λh. (f (x + h) - f x) / h) ─0→⇩N⇩S D"
then have deriv: "(λh. - ((f(x+h) - f x) / h)) ─0→⇩N⇩S - D"
by (rule NSLIM_minus)
have "∀h. - ((f (x + h) - f x) / h) = (- f (x + h) + f x) / h"
with deriv have "(λh. (- f (x + h) + f x) / h) ─0→⇩N⇩S - D"
by simp
then show "(λh. (f (x + h) - f x) / h) ─0→⇩N⇩S D ⟹ (λh. (f x - f (x + h)) / h) ─0→⇩N⇩S - D"
by simp
qed

text ‹Subtraction.›
"NSDERIV f x :> Da ⟹ NSDERIV g x :> Db ⟹ NSDERIV (λx. f x + - g x) x :> Da + - Db"

lemma NSDERIV_diff:
"NSDERIV f x :> Da ⟹ NSDERIV g x :> Db ⟹ NSDERIV (λx. f x - g x) x :> Da - Db"
using NSDERIV_add_minus [of f x Da g Db] by simp

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.›

subsection ‹Lemmas›

lemma NSDERIV_zero:
"⟦NSDERIV g x :> D; ( *f* g) (star_of x + y) = star_of (g x); y ∈ Infinitesimal; y ≠ 0⟧
⟹ D = 0"

text ‹Can be proved differently using ‹NSLIM_isCont_iff›.›
lemma NSDERIV_approx:
"NSDERIV f x :> D ⟹ h ∈ Infinitesimal ⟹ h ≠ 0 ⟹
( *f* f) (star_of x + h) - star_of (f x) ≈ 0"
by (meson DiffI Infinitesimal_ratio approx_star_of_HFinite mem_infmal_iff nsderiv_def singletonD)

text ‹From one version of differentiability

‹f x - f a›
‹-------------- ≈ Db›
‹x - a›
›

lemma NSDERIVD1:
"⟦NSDERIV f (g x) :> Da;
( *f* g) (star_of x + y) ≠ star_of (g x);
( *f* g) (star_of x + y) ≈ star_of (g x)⟧
⟹ (( *f* f) (( *f* g) (star_of x + y)) -
star_of (f (g x))) / (( *f* g) (star_of x + y) - star_of (g x)) ≈
star_of Da"
by (auto simp add: NSDERIV_NSLIM_iff2 NSLIM_def)

text ‹From other version of differentiability

‹f (x + h) - f x›
‹------------------ ≈ Db›
‹h›
›

lemma NSDERIVD2: "[| NSDERIV g x :> Db; y ∈ Infinitesimal; y ≠ 0 |]
==> (( *f* g) (star_of(x) + y) - star_of(g x)) / y
≈ star_of(Db)"
by (auto simp add: NSDERIV_NSLIM_iff NSLIM_def mem_infmal_iff starfun_lambda_cancel)

text ‹This proof uses both definitions of differentiability.›
lemma NSDERIV_chain:
"NSDERIV f (g x) :> Da ⟹ NSDERIV g x :> Db ⟹ NSDERIV (f ∘ g) x :> Da * Db"
using DERIV_NS_iff DERIV_chain NSDERIV_NSLIM_iff by blast

text ‹Differentiation of natural number powers.›
lemma NSDERIV_Id [simp]: "NSDERIV (λx. x) x :> 1"
by (simp add: NSDERIV_NSLIM_iff NSLIM_def del: divide_self_if)

lemma NSDERIV_cmult_Id [simp]: "NSDERIV ((*) c) x :> c"
using NSDERIV_Id [THEN NSDERIV_cmult] by simp

lemma NSDERIV_inverse:
fixes x :: "'a::real_normed_field"
assumes "x ≠ 0" ― ‹can't get rid of \<^term>‹x ≠ 0› because it isn't continuous at zero›
shows "NSDERIV (λx. inverse x) x :> - (inverse x ^ Suc (Suc 0))"
proof -
{
fix h :: "'a star"
assume h_Inf: "h ∈ Infinitesimal"
from this assms have not_0: "star_of x + h ≠ 0"
assume "h ≠ 0"
from h_Inf have "h * star_of x ∈ Infinitesimal"
by (rule Infinitesimal_HFinite_mult) simp
with assms have "inverse (- (h * star_of x) + - (star_of x * star_of x)) ≈
inverse (- (star_of x * star_of x))"
proof -
have "- (h * star_of x) + - (star_of x * star_of x) ≈ - (star_of x * star_of x)"
using ‹h * star_of x ∈ Infinitesimal› assms bex_Infinitesimal_iff by fastforce
then show ?thesis
by (metis assms mult_eq_0_iff neg_equal_0_iff_equal star_of_approx_inverse star_of_minus star_of_mult)
qed
moreover from not_0 ‹h ≠ 0› assms
have "inverse (- (h * star_of x) + - (star_of x * star_of x))
= (inverse (star_of x + h) - inverse (star_of x)) / h"
by (simp add: division_ring_inverse_diff inverse_mult_distrib [symmetric]
inverse_minus_eq [symmetric] algebra_simps)
ultimately have "(inverse (star_of x + h) - inverse (star_of x)) / h ≈
- (inverse (star_of x) * inverse (star_of x))"
using assms by simp
}
then show ?thesis by (simp add: nsderiv_def)
qed

subsubsection ‹Equivalence of NS and Standard definitions›

lemma divideR_eq_divide: "x /⇩R y = x / y"

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)

text ‹NS version.›
lemma NSDERIV_pow: "NSDERIV (λx. x ^ n) x :> real n * (x ^ (n - Suc 0))"

text ‹Derivative of inverse.›
lemma NSDERIV_inverse_fun:
"NSDERIV f x :> d ⟹ f x ≠ 0 ⟹
NSDERIV (λx. inverse (f x)) x :> (- (d * inverse (f x ^ Suc (Suc 0))))"
for x :: "'a::{real_normed_field}"
by (simp add: NSDERIV_DERIV_iff DERIV_inverse_fun del: power_Suc)

text ‹Derivative of quotient.›
lemma NSDERIV_quotient:
fixes x :: "'a::real_normed_field"
shows "NSDERIV f x :> d ⟹ NSDERIV g x :> e ⟹ g x ≠ 0 ⟹
NSDERIV (λy. 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: power_Suc)

lemma CARAT_NSDERIV:
"NSDERIV f x :> l ⟹ ∃g. (∀z. f z - f x = g z * (z - x)) ∧ isNSCont g x ∧ g x = l"
by (simp add: CARAT_DERIV NSDERIV_DERIV_iff isNSCont_isCont_iff)

lemma hypreal_eq_minus_iff3: "x = y + z ⟷ x + - z = y"
for x y z :: hypreal
by auto

lemma CARAT_DERIVD:
assumes all: "∀z. f z - f x = g z * (z - x)"
and nsc: "isNSCont g x"
shows "NSDERIV f x :> g x"
proof -
from nsc have "∀w. w ≠ star_of x ∧ w ≈ star_of x ⟶
( *f* g) w * (w - star_of x) / (w - star_of x) ≈ star_of (g x)"
with all show ?thesis
by (simp add: NSDERIV_iff2 starfun_if_eq cong: if_cong)
qed

subsubsection ‹Differentiability predicate›

lemma NSdifferentiableD: "f NSdifferentiable x ⟹ ∃D. NSDERIV f x :> D"

lemma NSdifferentiableI: "NSDERIV f x :> D ⟹ f NSdifferentiable x"

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)"
by (erule NSdifferentiableI [THEN incrementI])

text ‹The Increment theorem -- Keisler p. 65.›
lemma increment_thm:
assumes "NSDERIV f x :> D" "h ∈ Infinitesimal" "h ≠ 0"
shows "∃e ∈ Infinitesimal. increment f x h = hypreal_of_real D * h + e * h"
proof -
have inc: "increment f x h = ( *f* f) (hypreal_of_real x + h) - hypreal_of_real (f x)"
using assms(1) incrementI2 by auto
have "(( *f* f) (hypreal_of_real x + h) - hypreal_of_real (f x)) / h ≈ hypreal_of_real D"
then obtain y where "y ∈ Infinitesimal"
"(( *f* f) (hypreal_of_real x + h) - hypreal_of_real (f x)) / h = hypreal_of_real D + y"
by (metis bex_Infinitesimal_iff2)
then have "increment f x h / h = hypreal_of_real D + y"
by (metis inc)
then show ?thesis
by (metis (no_types) ‹y ∈ Infinitesimal› ‹h ≠ 0› distrib_right mult.commute nonzero_mult_div_cancel_left times_divide_eq_right)
qed

lemma increment_approx_zero: "NSDERIV f x :> D ⟹ h ≈ 0 ⟹ h ≠ 0 ⟹ increment f x h ≈ 0"
by (simp add: NSDERIV_approx incrementI2 mem_infmal_iff)

end
```