# Theory HyperDef

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

section ‹Construction of Hyperreals Using Ultrafilters›

theory HyperDef
imports Complex_Main HyperNat
begin

type_synonym hypreal = "real star"

abbreviation hypreal_of_real :: "real ⇒ real star"
where "hypreal_of_real ≡ star_of"

abbreviation hypreal_of_hypnat :: "hypnat ⇒ hypreal"
where "hypreal_of_hypnat ≡ of_hypnat"

definition omega :: hypreal  ("ω")
where "ω = star_n (λn. real (Suc n))"
― ‹an infinite number ‹= [<1, 2, 3, …>]››

definition epsilon :: hypreal  ("ε")
where "ε = star_n (λn. inverse (real (Suc n)))"
― ‹an infinitesimal number ‹= [<1, 1/2, 1/3, …>]››

subsection ‹Real vector class instances›

instantiation star :: (scaleR) scaleR
begin
definition star_scaleR_def [transfer_unfold]: "scaleR r ≡ *f* (scaleR r)"
instance ..
end

lemma Standard_scaleR [simp]: "x ∈ Standard ⟹ scaleR r x ∈ Standard"

lemma star_of_scaleR [simp]: "star_of (scaleR r x) = scaleR r (star_of x)"
by transfer (rule refl)

instance star :: (real_vector) real_vector
proof
fix a b :: real
show "⋀x y::'a star. scaleR a (x + y) = scaleR a x + scaleR a y"
by transfer (rule scaleR_right_distrib)
show "⋀x::'a star. scaleR (a + b) x = scaleR a x + scaleR b x"
by transfer (rule scaleR_left_distrib)
show "⋀x::'a star. scaleR a (scaleR b x) = scaleR (a * b) x"
by transfer (rule scaleR_scaleR)
show "⋀x::'a star. scaleR 1 x = x"
by transfer (rule scaleR_one)
qed

instance star :: (real_algebra) real_algebra
proof
fix a :: real
show "⋀x y::'a star. scaleR a x * y = scaleR a (x * y)"
by transfer (rule mult_scaleR_left)
show "⋀x y::'a star. x * scaleR a y = scaleR a (x * y)"
by transfer (rule mult_scaleR_right)
qed

instance star :: (real_algebra_1) real_algebra_1 ..

instance star :: (real_div_algebra) real_div_algebra ..

instance star :: (field_char_0) field_char_0 ..

instance star :: (real_field) real_field ..

lemma star_of_real_def [transfer_unfold]: "of_real r = star_of (of_real r)"
by (unfold of_real_def, transfer, rule refl)

lemma Standard_of_real [simp]: "of_real r ∈ Standard"

lemma star_of_of_real [simp]: "star_of (of_real r) = of_real r"
by transfer (rule refl)

lemma of_real_eq_star_of [simp]: "of_real = star_of"
proof
show "of_real r = star_of r" for r :: real
by transfer simp
qed

lemma Reals_eq_Standard: "(ℝ :: hypreal set) = Standard"

subsection ‹Injection from @{typ hypreal}›

definition of_hypreal :: "hypreal ⇒ 'a::real_algebra_1 star"
where [transfer_unfold]: "of_hypreal = *f* of_real"

lemma Standard_of_hypreal [simp]: "r ∈ Standard ⟹ of_hypreal r ∈ Standard"

lemma of_hypreal_0 [simp]: "of_hypreal 0 = 0"
by transfer (rule of_real_0)

lemma of_hypreal_1 [simp]: "of_hypreal 1 = 1"
by transfer (rule of_real_1)

lemma of_hypreal_add [simp]: "⋀x y. of_hypreal (x + y) = of_hypreal x + of_hypreal y"

lemma of_hypreal_minus [simp]: "⋀x. of_hypreal (- x) = - of_hypreal x"
by transfer (rule of_real_minus)

lemma of_hypreal_diff [simp]: "⋀x y. of_hypreal (x - y) = of_hypreal x - of_hypreal y"
by transfer (rule of_real_diff)

lemma of_hypreal_mult [simp]: "⋀x y. of_hypreal (x * y) = of_hypreal x * of_hypreal y"
by transfer (rule of_real_mult)

lemma of_hypreal_inverse [simp]:
"⋀x. of_hypreal (inverse x) =
inverse (of_hypreal x :: 'a::{real_div_algebra, division_ring} star)"
by transfer (rule of_real_inverse)

lemma of_hypreal_divide [simp]:
"⋀x y. of_hypreal (x / y) =
(of_hypreal x / of_hypreal y :: 'a::{real_field, field} star)"
by transfer (rule of_real_divide)

lemma of_hypreal_eq_iff [simp]: "⋀x y. (of_hypreal x = of_hypreal y) = (x = y)"
by transfer (rule of_real_eq_iff)

lemma of_hypreal_eq_0_iff [simp]: "⋀x. (of_hypreal x = 0) = (x = 0)"
by transfer (rule of_real_eq_0_iff)

subsection ‹Properties of @{term starrel}›

lemma lemma_starrel_refl [simp]: "x ∈ starrel `` {x}"

lemma starrel_in_hypreal [simp]: "starrel``{x}:star"
by (simp add: star_def starrel_def quotient_def, blast)

declare Abs_star_inject [simp] Abs_star_inverse [simp]
declare equiv_starrel [THEN eq_equiv_class_iff, simp]

subsection ‹@{term hypreal_of_real}: the Injection from @{typ real} to @{typ hypreal}›

lemma inj_star_of: "inj star_of"
by (rule inj_onI) simp

lemma mem_Rep_star_iff: "X ∈ Rep_star x ⟷ x = star_n X"
by (cases x) (simp add: star_n_def)

lemma Rep_star_star_n_iff [simp]: "X ∈ Rep_star (star_n Y) ⟷ eventually (λn. Y n = X n) 𝒰"

lemma Rep_star_star_n: "X ∈ Rep_star (star_n X)"
by simp

subsection ‹Properties of @{term star_n}›

lemma star_n_add: "star_n X + star_n Y = star_n (λn. X n + Y n)"

lemma star_n_minus: "- star_n X = star_n (λn. -(X n))"
by (simp only: star_minus_def starfun_star_n)

lemma star_n_diff: "star_n X - star_n Y = star_n (λn. X n - Y n)"
by (simp only: star_diff_def starfun2_star_n)

lemma star_n_mult: "star_n X * star_n Y = star_n (λn. X n * Y n)"
by (simp only: star_mult_def starfun2_star_n)

lemma star_n_inverse: "inverse (star_n X) = star_n (λn. inverse (X n))"
by (simp only: star_inverse_def starfun_star_n)

lemma star_n_le: "star_n X ≤ star_n Y = eventually (λn. X n ≤ Y n) 𝒰"
by (simp only: star_le_def starP2_star_n)

lemma star_n_less: "star_n X < star_n Y = eventually (λn. X n < Y n) 𝒰"
by (simp only: star_less_def starP2_star_n)

lemma star_n_zero_num: "0 = star_n (λn. 0)"
by (simp only: star_zero_def star_of_def)

lemma star_n_one_num: "1 = star_n (λn. 1)"
by (simp only: star_one_def star_of_def)

lemma star_n_abs: "¦star_n X¦ = star_n (λn. ¦X n¦)"
by (simp only: star_abs_def starfun_star_n)

lemma hypreal_omega_gt_zero [simp]: "0 < ω"
by (simp add: omega_def star_n_zero_num star_n_less)

subsection ‹Existence of Infinite Hyperreal Number›

text ‹Existence of infinite number not corresponding to any real number.
Use assumption that member @{term 𝒰} is not finite.›

text ‹A few lemmas first.›

lemma lemma_omega_empty_singleton_disj:
"{n::nat. x = real n} = {} ∨ (∃y. {n::nat. x = real n} = {y})"
by force

lemma lemma_finite_omega_set: "finite {n::nat. x = real n}"
using lemma_omega_empty_singleton_disj [of x] by auto

lemma not_ex_hypreal_of_real_eq_omega: "∄x. hypreal_of_real x = ω"
apply (simp add: omega_def star_of_def star_n_eq_iff)
apply clarify
apply (rule_tac x2="x-1" in lemma_finite_omega_set [THEN FreeUltrafilterNat.finite, THEN notE])
apply (erule eventually_mono)
apply auto
done

lemma hypreal_of_real_not_eq_omega: "hypreal_of_real x ≠ ω"
using not_ex_hypreal_of_real_eq_omega by auto

text ‹Existence of infinitesimal number also not corresponding to any real number.›

lemma lemma_epsilon_empty_singleton_disj:
"{n::nat. x = inverse(real(Suc n))} = {} ∨ (∃y. {n::nat. x = inverse(real(Suc n))} = {y})"
by auto

lemma lemma_finite_epsilon_set: "finite {n. x = inverse (real (Suc n))}"
using lemma_epsilon_empty_singleton_disj [of x] by auto

lemma not_ex_hypreal_of_real_eq_epsilon: "∄x. hypreal_of_real x = ε"
by (auto simp: epsilon_def star_of_def star_n_eq_iff
lemma_finite_epsilon_set [THEN FreeUltrafilterNat.finite] simp del: of_nat_Suc)

lemma hypreal_of_real_not_eq_epsilon: "hypreal_of_real x ≠ ε"
using not_ex_hypreal_of_real_eq_epsilon by auto

lemma hypreal_epsilon_not_zero: "ε ≠ 0"
by (simp add: epsilon_def star_zero_def star_of_def star_n_eq_iff FreeUltrafilterNat.proper
del: star_of_zero)

lemma hypreal_epsilon_inverse_omega: "ε = inverse ω"
by (simp add: epsilon_def omega_def star_n_inverse)

lemma hypreal_epsilon_gt_zero: "0 < ε"

subsection ‹Absolute Value Function for the Hyperreals›

lemma hrabs_add_less: "¦x¦ < r ⟹ ¦y¦ < s ⟹ ¦x + y¦ < r + s"
for x y r s :: hypreal
by (simp add: abs_if split: if_split_asm)

lemma hrabs_less_gt_zero: "¦x¦ < r ⟹ 0 < r"
for x r :: hypreal
by (blast intro!: order_le_less_trans abs_ge_zero)

lemma hrabs_disj: "¦x¦ = x ∨ ¦x¦ = -x"
for x :: "'a::abs_if"

lemma hrabs_add_lemma_disj: "y + - x + (y + - z) = ¦x + - z¦ ⟹ y = z ∨ x = y"
for x y z :: hypreal
by (simp add: abs_if split: if_split_asm)

subsection ‹Embedding the Naturals into the Hyperreals›

abbreviation hypreal_of_nat :: "nat ⇒ hypreal"
where "hypreal_of_nat ≡ of_nat"

lemma SNat_eq: "Nats = {n. ∃N. n = hypreal_of_nat N}"

text ‹Naturals embedded in hyperreals: is a hyperreal c.f. NS extension.›

lemma hypreal_of_nat: "hypreal_of_nat m = star_n (λn. real m)"

declaration ‹
K (Lin_Arith.add_inj_thms [@{thm star_of_le} RS iffD2,
@{thm star_of_less} RS iffD2, @{thm star_of_eq} RS iffD2]
#> Lin_Arith.add_simps [@{thm star_of_zero}, @{thm star_of_one},
@{thm star_of_minus}, @{thm star_of_diff}, @{thm star_of_mult}]
#> Lin_Arith.add_inj_const (@{const_name "StarDef.star_of"}, @{typ "real ⇒ hypreal"}))
›

simproc_setup fast_arith_hypreal ("(m::hypreal) < n" | "(m::hypreal) ≤ n" | "(m::hypreal) = n") =
‹K Lin_Arith.simproc›

subsection ‹Exponentials on the Hyperreals›

lemma hpowr_0 [simp]: "r ^ 0 = (1::hypreal)"
for r :: hypreal
by (rule power_0)

lemma hpowr_Suc [simp]: "r ^ (Suc n) = r * (r ^ n)"
for r :: hypreal
by (rule power_Suc)

lemma hrealpow_two: "r ^ Suc (Suc 0) = r * r"
for r :: hypreal
by simp

lemma hrealpow_two_le [simp]: "0 ≤ r ^ Suc (Suc 0)"
for r :: hypreal

lemma hrealpow_two_le_add_order [simp]: "0 ≤ u ^ Suc (Suc 0) + v ^ Suc (Suc 0)"
for u v :: hypreal

lemma hrealpow_two_le_add_order2 [simp]: "0 ≤ u ^ Suc (Suc 0) + v ^ Suc (Suc 0) + w ^ Suc (Suc 0)"
for u v w :: hypreal

lemma hypreal_add_nonneg_eq_0_iff: "0 ≤ x ⟹ 0 ≤ y ⟹ x + y = 0 ⟷ x = 0 ∧ y = 0"
for x y :: hypreal
by arith

(* FIXME: DELETE THESE *)
lemma hypreal_three_squares_add_zero_iff: "x * x + y * y + z * z = 0 ⟷ x = 0 ∧ y = 0 ∧ z = 0"
for x y z :: hypreal

"x ^ Suc (Suc 0) + y ^ Suc (Suc 0) + z ^ Suc (Suc 0) = 0 ⟷ x = 0 ∧ y = 0 ∧ z = 0"
for x y z :: hypreal

(*FIXME: This and RealPow.abs_realpow_two should be replaced by an abstract
result proved in Rings or Fields*)
lemma hrabs_hrealpow_two [simp]: "¦x ^ Suc (Suc 0)¦ = x ^ Suc (Suc 0)"
for x :: hypreal

lemma two_hrealpow_ge_one [simp]: "(1::hypreal) ≤ 2 ^ n"
using power_increasing [of 0 n "2::hypreal"] by simp

lemma hrealpow: "star_n X ^ m = star_n (λn. (X n::real) ^ m)"
by (induct m) (auto simp: star_n_one_num star_n_mult)

lemma hrealpow_sum_square_expand:
"(x + y) ^ Suc (Suc 0) =
x ^ Suc (Suc 0) + y ^ Suc (Suc 0) + (hypreal_of_nat (Suc (Suc 0))) * x * y"
for x y :: hypreal

lemma power_hypreal_of_real_numeral:
"(numeral v :: hypreal) ^ n = hypreal_of_real ((numeral v) ^ n)"
by simp
declare power_hypreal_of_real_numeral [of _ "numeral w", simp] for w

lemma power_hypreal_of_real_neg_numeral:
"(- numeral v :: hypreal) ^ n = hypreal_of_real ((- numeral v) ^ n)"
by simp
declare power_hypreal_of_real_neg_numeral [of _ "numeral w", simp] for w
(*
lemma hrealpow_HFinite:
fixes x :: "'a::{real_normed_algebra,power} star"
shows "x ∈ HFinite ==> x ^ n ∈ HFinite"
apply (induct_tac "n")
apply (auto simp add: power_Suc intro: HFinite_mult)
done
*)

subsection ‹Powers with Hypernatural Exponents›

text ‹Hypernatural powers of hyperreals.›
definition pow :: "'a::power star ⇒ nat star ⇒ 'a star"  (infixr "pow" 80)
where hyperpow_def [transfer_unfold]: "R pow N = ( *f2* op ^) R N"

lemma Standard_hyperpow [simp]: "r ∈ Standard ⟹ n ∈ Standard ⟹ r pow n ∈ Standard"

lemma hyperpow: "star_n X pow star_n Y = star_n (λn. X n ^ Y n)"

lemma hyperpow_zero [simp]: "⋀n. (0::'a::{power,semiring_0} star) pow (n + (1::hypnat)) = 0"
by transfer simp

lemma hyperpow_not_zero: "⋀r n. r ≠ (0::'a::{field} star) ⟹ r pow n ≠ 0"
by transfer (rule power_not_zero)

lemma hyperpow_inverse: "⋀r n. r ≠ (0::'a::field star) ⟹ inverse (r pow n) = (inverse r) pow n"
by transfer (rule power_inverse [symmetric])

lemma hyperpow_hrabs: "⋀r n. ¦r::'a::{linordered_idom} star¦ pow n = ¦r pow n¦"
by transfer (rule power_abs [symmetric])

lemma hyperpow_add: "⋀r n m. (r::'a::monoid_mult star) pow (n + m) = (r pow n) * (r pow m)"

lemma hyperpow_one [simp]: "⋀r. (r::'a::monoid_mult star) pow (1::hypnat) = r"
by transfer (rule power_one_right)

lemma hyperpow_two: "⋀r. (r::'a::monoid_mult star) pow (2::hypnat) = r * r"
by transfer (rule power2_eq_square)

lemma hyperpow_gt_zero: "⋀r n. (0::'a::{linordered_semidom} star) < r ⟹ 0 < r pow n"
by transfer (rule zero_less_power)

lemma hyperpow_ge_zero: "⋀r n. (0::'a::{linordered_semidom} star) ≤ r ⟹ 0 ≤ r pow n"
by transfer (rule zero_le_power)

lemma hyperpow_le: "⋀x y n. (0::'a::{linordered_semidom} star) < x ⟹ x ≤ y ⟹ x pow n ≤ y pow n"
by transfer (rule power_mono [OF _ order_less_imp_le])

lemma hyperpow_eq_one [simp]: "⋀n. 1 pow n = (1::'a::monoid_mult star)"
by transfer (rule power_one)

lemma hrabs_hyperpow_minus [simp]: "⋀(a::'a::linordered_idom star) n. ¦(-a) pow n¦ = ¦a pow n¦"
by transfer (rule abs_power_minus)

lemma hyperpow_mult: "⋀r s n. (r * s::'a::comm_monoid_mult star) pow n = (r pow n) * (s pow n)"
by transfer (rule power_mult_distrib)

lemma hyperpow_two_le [simp]: "⋀r. (0::'a::{monoid_mult,linordered_ring_strict} star) ≤ r pow 2"
by (auto simp add: hyperpow_two zero_le_mult_iff)

lemma hrabs_hyperpow_two [simp]:
"¦x pow 2¦ = (x::'a::{monoid_mult,linordered_ring_strict} star) pow 2"
by (simp only: abs_of_nonneg hyperpow_two_le)

lemma hyperpow_two_hrabs [simp]: "¦x::'a::linordered_idom star¦ pow 2 = x pow 2"

text ‹The precondition could be weakened to @{term "0≤x"}.›
lemma hypreal_mult_less_mono: "u < v ⟹ x < y ⟹ 0 < v ⟹ 0 < x ⟹ u * x < v * y"
for u v x y :: hypreal

lemma hyperpow_two_gt_one: "⋀r::'a::linordered_semidom star. 1 < r ⟹ 1 < r pow 2"
by transfer simp

lemma hyperpow_two_ge_one: "⋀r::'a::linordered_semidom star. 1 ≤ r ⟹ 1 ≤ r pow 2"
by transfer (rule one_le_power)

lemma two_hyperpow_ge_one [simp]: "(1::hypreal) ≤ 2 pow n"
apply (rule_tac y = "1 pow n" in order_trans)
apply (rule_tac [2] hyperpow_le)
apply auto
done

lemma hyperpow_minus_one2 [simp]: "⋀n. (- 1) pow (2 * n) = (1::hypreal)"
by transfer (rule power_minus1_even)

lemma hyperpow_less_le: "⋀r n N. (0::hypreal) ≤ r ⟹ r ≤ 1 ⟹ n < N ⟹ r pow N ≤ r pow n"
by transfer (rule power_decreasing [OF order_less_imp_le])

lemma hyperpow_SHNat_le:
"0 ≤ r ⟹ r ≤ (1::hypreal) ⟹ N ∈ HNatInfinite ⟹ ∀n∈Nats. r pow N ≤ r pow n"
by (auto intro!: hyperpow_less_le simp: HNatInfinite_iff)

lemma hyperpow_realpow: "(hypreal_of_real r) pow (hypnat_of_nat n) = hypreal_of_real (r ^ n)"
by transfer (rule refl)

lemma hyperpow_SReal [simp]: "(hypreal_of_real r) pow (hypnat_of_nat n) ∈ ℝ"

lemma hyperpow_zero_HNatInfinite [simp]: "N ∈ HNatInfinite ⟹ (0::hypreal) pow N = 0"
by (drule HNatInfinite_is_Suc, auto)

lemma hyperpow_le_le: "(0::hypreal) ≤ r ⟹ r ≤ 1 ⟹ n ≤ N ⟹ r pow N ≤ r pow n"
apply (drule order_le_less [of n, THEN iffD1])
apply (auto intro: hyperpow_less_le)
done

lemma hyperpow_Suc_le_self2: "(0::hypreal) ≤ r ⟹ r < 1 ⟹ r pow (n + (1::hypnat)) ≤ r"
apply (drule_tac n = " (1::hypnat) " in hyperpow_le_le)
apply auto
done

lemma hyperpow_hypnat_of_nat: "⋀x. x pow hypnat_of_nat n = x ^ n"
by transfer (rule refl)

lemma of_hypreal_hyperpow:
"⋀x n. of_hypreal (x pow n) = (of_hypreal x::'a::{real_algebra_1} star) pow n"
by transfer (rule of_real_power)

end
```