Theory HyperDef

(*  Title:      HOL/Nonstandard_Analysis/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

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
  definition star_scaleR_def [transfer_unfold]: "scaleR r  *f* (scaleR r)"
  instance ..

lemma Standard_scaleR [simp]: "x  Standard  scaleR r x  Standard"
  by (simp add: star_scaleR_def)

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
  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)

instance star :: (real_algebra) real_algebra
  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)

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"
  by (simp add: star_of_real_def)

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"
  show "of_real r = star_of r" for r :: real
    by transfer simp

lemma Reals_eq_Standard: "( :: hypreal set) = Standard"
  by (simp add: Reals_def Standard_def)

subsection Injection from typhypreal

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"
  by (simp add: of_hypreal_def)

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"
  by transfer (rule of_real_add)

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 termstarrel

lemma lemma_starrel_refl [simp]: "x  starrel `` {x}"
  by (simp add: starrel_def)

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 termhypreal_of_real: the Injection from typreal to typhypreal

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"
  by (simp add: star_n_def)

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

subsection Properties of termstar_n

lemma star_n_add: "star_n X + star_n Y = star_n (λn. X n + Y n)"
  by (simp only: star_add_def starfun2_star_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 term𝒰 is not finite.

lemma hypreal_of_real_not_eq_omega: "hypreal_of_real x  ω"
proof -
  have False if "F n in 𝒰. x = 1 + real n" for x
  proof -
    have "finite {n::nat. x = 1 + real n}"
      by (simp add: finite_nat_set_iff_bounded_le) (metis add.commute nat_le_linear nat_le_real_less)
    then show False
      using FreeUltrafilterNat.finite that by blast
  then show ?thesis
    by (auto simp add: omega_def star_of_def star_n_eq_iff)

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

lemma hypreal_of_real_not_eq_epsilon: "hypreal_of_real x  ε"
proof -
  have False if "F n in 𝒰. x = inverse (1 + real n)" for x
  proof -
    have "finite {n::nat. x = inverse (1 + real n)}"
      by (simp add: finite_nat_set_iff_bounded_le) (metis add.commute inverse_inverse_eq linear nat_le_real_less of_nat_Suc) 
    then show False
      using FreeUltrafilterNat.finite that by blast
  then show ?thesis
    by (auto simp: epsilon_def star_of_def star_n_eq_iff)

lemma epsilon_ge_zero [simp]: "0  ε"
  by (simp add: epsilon_def star_n_zero_num star_n_le)

lemma epsilon_not_zero: "ε  0"
  using hypreal_of_real_not_eq_epsilon by force

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

lemma epsilon_gt_zero: "0 < ε"
  by (simp add: epsilon_inverse_omega)

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}"
  by (simp add: Nats_def image_def)

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)"
  by (simp add: star_of_def [symmetric])

  K (Lin_Arith.add_simps @{thms star_of_zero star_of_one
      star_of_numeral star_of_add
      star_of_minus star_of_diff star_of_mult}
  #> Lin_Arith.add_inj_thms @{thms star_of_le [THEN iffD2]
      star_of_less [THEN iffD2] star_of_eq [THEN iffD2]}
  #> Lin_Arith.add_inj_const (const_nameStarDef.star_of, typreal  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: "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
  by (simp add: distrib_left distrib_right)

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

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* (^)) R N"

lemma Standard_hyperpow [simp]: "r  Standard  n  Standard  r pow n  Standard"
  by (simp add: hyperpow_def)

lemma hyperpow: "star_n X pow star_n Y = star_n (λn. X n ^ Y n)"
  by (simp add: hyperpow_def starfun2_star_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)"
  by transfer (rule power_add)

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 hyperpow_two_hrabs [simp]: "¦x::'a::linordered_idom star¦ pow 2 = x pow 2"
  by (simp add: hyperpow_hrabs)

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"
  by (metis hyperpow_eq_one hyperpow_le one_le_numeral zero_less_one)

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  nNats. 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)  "
  by (simp add: Reals_eq_Standard)

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"
  by (metis hyperpow_less_le le_less)

lemma hyperpow_Suc_le_self2: "(0::hypreal)  r  r < 1  r pow (n + (1::hypnat))  r"
  by (metis hyperpow_less_le hyperpow_one hypnat_add_self_le le_less)

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)