# Theory HyperDef

theory HyperDef
imports Complex_Main HyperNat
```(*  Title       : HOL/NSA/HyperDef.thy
Author      : Jacques D. Fleuriot
Copyright   : 1998  University of Cambridge
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
― ‹an infinite number ‹= [<1,2,3,...>]››
"ω = star_n (λn. real (Suc n))"

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

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
fix r :: real
show "of_real r = star_of r"
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)

"⋀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}›

"star_n X + star_n Y = star_n (%n. X n + Y n)"

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

lemma star_n_diff:
"star_n X - star_n Y = star_n (%n. 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 (%n. X n * Y n)"
by (simp only: star_mult_def starfun2_star_n)

lemma star_n_inverse:
"inverse (star_n X) = star_n (%n. 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) FreeUltrafilterNat)"
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) FreeUltrafilterNat)"
by (simp only: star_less_def starP2_star_n)

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

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

lemma star_n_abs: "¦star_n X¦ = star_n (%n. ¦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 FreeUltrafilterNat} 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"
by (insert not_ex_hypreal_of_real_eq_omega, 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))}"
by (cut_tac x = x in lemma_epsilon_empty_singleton_disj, auto)

lemma not_ex_hypreal_of_real_eq_epsilon: "~ (∃x. hypreal_of_real x = ε)"
by (auto simp add: 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"
by (insert not_ex_hypreal_of_real_eq_epsilon, 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::hypreal)"
by (simp add: abs_if split: split_if_asm)

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

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

lemma hrabs_add_lemma_disj: "(y::hypreal) + - x + (y + - z) = ¦x + - z¦ ==> y = z | x = y"

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}"

(*------------------------------------------------------------*)
(* naturals embedded in hyperreals                            *)
(* is a hyperreal c.f. NS extension                           *)
(*------------------------------------------------------------*)

lemma hypreal_of_nat: "hypreal_of_nat m = star_n (%n. 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)"
by (rule power_0)

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

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

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

"(0::hypreal) ≤ u ^ Suc (Suc 0) + v ^ Suc (Suc 0)"

"(0::hypreal) ≤ u ^ Suc (Suc 0) + v ^ Suc (Suc 0) + w ^ Suc (Suc 0)"

"[| 0 ≤ x; 0 ≤ y |] ==> (x+y = 0) = (x = 0 & y = (0::hypreal))"
by arith

text‹FIXME: DELETE THESE›
"(x*x + y*y + z*z = 0) = (x = 0 & y = 0 & z = (0::hypreal))"
done

"(x ^ Suc (Suc 0) + y ^ Suc (Suc 0) + z ^ Suc (Suc 0) = (0::hypreal)) =
(x = 0 & y = 0 & z = 0)"

(*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::hypreal) ^ Suc (Suc 0)"

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

lemma hrealpow:
"star_n X ^ m = star_n (%n. (X n::real) ^ m)"
apply (induct_tac "m")
apply (auto simp add: star_n_one_num star_n_mult power_0)
done

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

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›

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

lemma Standard_hyperpow [simp]:
"⟦r ∈ Standard; n ∈ Standard⟧ ⟹ r pow n ∈ Standard"
unfolding hyperpow_def by simp

lemma hyperpow: "star_n X pow star_n Y = star_n (%n. 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])

"⋀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::hypreal) < v;  0 < x |] ==> u*x < v* y"

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, 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 |]
==> ALL n: Nats. r pow N ≤ r pow n"
by (auto intro!: hyperpow_less_le simp add: 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
```