Quelle  HSeries.thy

(*  Title:      HOL/Nonstandard_Analysis/HSeries.thy
  Author: Jacques D. Fleuriot
  Copyright: 1998 University of Cambridge
 
 Converted to Isar and polished by lcp
*)

section ‹Finite Summation and Infinite Series for Hyperreals›

theory HSeries
  imports HSEQ
begin

definition sumhr :: "hypnat × hypnat × (nat ==> real) ==> hypreal"
  where "sumhr = (λ(M,N,f). starfun2 (λm n. sum f {m..

definition NSsums :: "(nat ==> real) ==> real ==> bool"  (infixr ‹NSsums› 80)
  where "f NSsums s = (λn. sum f {..<----🪙N🪙S s"

definition NSsummable :: "(nat ==> real) ==> bool"
  where "NSsummable f ⟷ (∃s. f NSsums s)"

definition NSsuminf :: "(nat ==> real) ==> real"
  where "NSsuminf f = (THE s. f NSsums s)"

lemma sumhr_app: "sumhr (M, N, f) = ( *f2* (λm n. sum f {m..
  by (simp add: sumhr_def)

text ‹Base case in definition of 🍋‹sumr›.›
lemma sumhr_zero [simp]: "∧m. sumhr (m, 0, f) = 0"
  unfolding sumhr_app by transfer simp

text ‹Recursive case in definition of 🍋‹sumr›.›
lemma sumhr_if:
  "∧m n. sumhr (m, n + 1, f) = (if n + 1 ≤ m then 0 else sumhr (m, n, f) + ( *f* f) n)"
  unfolding sumhr_app by transfer simp

lemma sumhr_Suc_zero [simp]: "∧n. sumhr (n + 1, n, f) = 0"
  unfolding sumhr_app by transfer simp

lemma sumhr_eq_bounds [simp]: "∧n. sumhr (n, n, f) = 0"
  unfolding sumhr_app by transfer simp

lemma sumhr_Suc [simp]: "∧m. sumhr (m, m + 1, f) = ( *f* f) m"
  unfolding sumhr_app by transfer simp

lemma sumhr_add_lbound_zero [simp]: "∧k m. sumhr (m + k, k, f) = 0"
  unfolding sumhr_app by transfer simp

lemma sumhr_add: "∧m n. sumhr (m, n, f) + sumhr (m, n, g) = sumhr (m, n, λi. f i + g i)"
  unfolding sumhr_app by transfer (rule sum.distrib [symmetric])

lemma sumhr_mult: "∧m n. hypreal_of_real r * sumhr (m, n, f) = sumhr (m, n, λn. r * f n)"
  unfolding sumhr_app by transfer (rule sum_distrib_left)

lemma sumhr_split_add: "∧n p. n < p ==> sumhr (0, n, f) + sumhr (n, p, f) = sumhr (0, p, f)"
  unfolding sumhr_app by transfer (simp add: sum.atLeastLessThan_concat)

lemma sumhr_split_diff: "n < p ==> sumhr (0, p, f) - sumhr (0, n, f) = sumhr (n, p, f)"
  by (drule sumhr_split_add [symmetric, where f = f]) simp

lemma sumhr_hrabs: "∧m n. ∣sumhr (m, n, f)∣ ≤ sumhr (m, n, λi. ∣f i∣)"
  unfolding sumhr_app by transfer (rule sum_abs)

text ‹Other general version also needed.›
lemma sumhr_fun_hypnat_eq:
  "(∀r. m ≤ r ∧ r < n ⟶ f r = g r) ⟶
    sumhr (hypnat_of_nat m, hypnat_of_nat n, f) =
    sumhr (hypnat_of_nat m, hypnat_of_nat n, g)"
  unfolding sumhr_app by transfer simp

lemma sumhr_const: "∧n. sumhr (0, n, λi. r) = hypreal_of_hypnat n * hypreal_of_real r"
  unfolding sumhr_app by transfer simp

lemma sumhr_less_bounds_zero [simp]: "∧m n. n < m ==> sumhr (m, n, f) = 0"
  unfolding sumhr_app by transfer simp

lemma sumhr_minus: "∧m n. sumhr (m, n, λi. - f i) = - sumhr (m, n, f)"
  unfolding sumhr_app by transfer (rule sum_negf)

lemma sumhr_shift_bounds:
  "∧m n. sumhr (m + hypnat_of_nat k, n + hypnat_of_nat k, f) =
    sumhr (m, n, λi. f (i + k))"
  unfolding sumhr_app by transfer (rule sum.shift_bounds_nat_ivl)


subsection ‹Nonstandard Sums›

text ‹Infinite sums are obtained by summing to some infinite hypernatural
  (such as 🍋‹whn›).›
lemma sumhr_hypreal_of_hypnat_omega: "sumhr (0, whn, λi. 1) = hypreal_of_hypnat whn"
  by (simp add: sumhr_const)

lemma whn_eq_ψm1: "hypreal_of_hypnat whn = ψ - 1"
  unfolding star_class_defs omega_def hypnat_omega_def of_hypnat_def star_of_def
  by (simp add: starfun_star_n starfun2_star_n)

lemma sumhr_hypreal_omega_minus_one: "sumhr(0, whn, λi. 1) = ψ - 1"
  by (simp add: sumhr_const whn_eq_ψm1)

lemma sumhr_minus_one_realpow_zero [simp]: "∧N. sumhr (0, N + N, λi. (-1) ^ (i + 1)) = 0"
  unfolding sumhr_app
  by transfer (induct_tac N, auto)

lemma sumhr_interval_const:
  "(∀n. m ≤ Suc n ⟶ f n = r) ∧ m ≤ na ==>
    sumhr (hypnat_of_nat m, hypnat_of_nat na, f) = hypreal_of_nat (na - m) * hypreal_of_real r"
  unfolding sumhr_app by transfer simp

lemma starfunNat_sumr: "∧N. ( *f* (λn. sum f {0..
  unfolding sumhr_app by transfer (rule refl)

lemma sumhr_hrabs_approx [simp]: "sumhr (0, M, f) ≈ sumhr (0, N, f) ==> ∣sumhr (M, N, f)∣ ≈ 0"
  using linorder_less_linear [where x = M and y = N]
  by (metis (no_types, lifting) abs_zero approx_hrabs approx_minus_iff approx_refl approx_sym sumhr_eq_bounds sumhr_less_bounds_zero sumhr_split_diff)


subsection ‹Infinite sums: Standard and NS theorems›

lemma sums_NSsums_iff: "f sums l ⟷ f NSsums l"
  by (simp add: sums_def NSsums_def LIMSEQ_NSLIMSEQ_iff)

lemma summable_NSsummable_iff: "summable f ⟷ NSsummable f"
  by (simp add: summable_def NSsummable_def sums_NSsums_iff)

lemma suminf_NSsuminf_iff: "suminf f = NSsuminf f"
  by (simp add: suminf_def NSsuminf_def sums_NSsums_iff)

lemma NSsums_NSsummable: "f NSsums l ==> NSsummable f"
  unfolding NSsums_def NSsummable_def by blast

lemma NSsummable_NSsums: "NSsummable f ==> f NSsums (NSsuminf f)"
  unfolding NSsummable_def NSsuminf_def NSsums_def
  by (blast intro: theI NSLIMSEQ_unique)

lemma NSsums_unique: "f NSsums s ==> s = NSsuminf f"
  by (simp add: suminf_NSsuminf_iff [symmetric] sums_NSsums_iff sums_unique)

lemma NSseries_zero: "∀m. n ≤ Suc m ⟶ f m = 0 ==> f NSsums (sum f {..
  by (auto simp add: sums_NSsums_iff [symmetric] not_le[symmetric] intro!: sums_finite)

lemma NSsummable_NSCauchy:
  "NSsummable f ⟷ (∀M ∈ HNatInfinite. ∀N ∈ HNatInfinite. ∣sumhr (M, N, f)∣ ≈ 0)" (is "?L=?R")
proof -
  have "?L = (∀M∈HNatInfinite. ∀N∈HNatInfinite. sumhr (0, M, f) ≈ sumhr (0, N, f))"
    by (auto simp add: summable_iff_convergent convergent_NSconvergent_iff NSCauchy_def starfunNat_sumr 
        simp flip: NSCauchy_NSconvergent_iff summable_NSsummable_iff atLeast0LessThan)
  also have "... ⟷ ?R"
    by (metis approx_hrabs_zero_cancel approx_minus_iff approx_refl approx_sym linorder_less_linear sumhr_hrabs_approx sumhr_split_diff)
  finally show ?thesis .
qed

text ‹Terms of a convergent series tend to zero.›
lemma NSsummable_NSLIMSEQ_zero: "NSsummable f ==> f <----🪙N🪙S 0"
  by (metis HNatInfinite_add NSLIMSEQ_def NSsummable_NSCauchy approx_hrabs_zero_cancel star_of_zero sumhr_Suc)

text ‹Nonstandard comparison test.›
lemma NSsummable_comparison_test: "∃N. ∀n. N ≤ n ⟶ ∣f n∣ ≤ g n ==> NSsummable g ==> NSsummable f"
  by (metis real_norm_def summable_NSsummable_iff summable_comparison_test)

lemma NSsummable_rabs_comparison_test:
  "∃N. ∀n. N ≤ n ⟶ ∣f n∣ ≤ g n ==> NSsummable g ==> NSsummable (λk. ∣f k∣)"
  by (rule NSsummable_comparison_test) auto

end