(* Title : HSeries.thy
Author : Jacques D. Fleuriot
Copyright : 1998 University of Cambridge
Converted to Isar and polished by lcp
*)
header{*Finite Summation and Infinite Series for Hyperreals*}
theory HSeries
imports Series
begin
definition
sumhr :: "(hypnat * hypnat * (nat=>real)) => hypreal" where
"sumhr =
(%(M,N,f). starfun2 (%m n. setsum f {m..<n}) M N)"
definition
NSsums :: "[nat=>real,real] => bool" (infixr "NSsums" 80) where
"f NSsums s = (%n. setsum f {0..<n}) ----NS> s"
definition
NSsummable :: "(nat=>real) => bool" where
"NSsummable f = (\<exists>s. f NSsums s)"
definition
NSsuminf :: "(nat=>real) => real" where
"NSsuminf f = (THE s. f NSsums s)"
lemma sumhr:
"sumhr(star_n M, star_n N, f) =
star_n (%n. setsum f {M n..<N n})"
by (simp add: sumhr_def starfun2_star_n)
text{*Base case in definition of @{term sumr}*}
lemma sumhr_zero [simp]: "sumhr (m,0,f) = 0"
apply (cases m)
apply (simp add: star_n_zero_num sumhr symmetric)
done
text{*Recursive case in definition of @{term sumr}*}
lemma sumhr_if:
"sumhr(m,n+1,f) =
(if n + 1 \<le> m then 0 else sumhr(m,n,f) + ( *f* f) n)"
apply (cases m, cases n)
apply (auto simp add: star_n_one_num sumhr star_n_add star_n_le starfun
star_n_zero_num star_n_eq_iff, ultra+)
done
lemma sumhr_Suc_zero [simp]: "sumhr (n + 1, n, f) = 0"
apply (cases n)
apply (simp add: star_n_one_num sumhr star_n_add star_n_zero_num)
done
lemma sumhr_eq_bounds [simp]: "sumhr (n,n,f) = 0"
apply (cases n)
apply (simp add: sumhr star_n_zero_num)
done
lemma sumhr_Suc [simp]: "sumhr (m,m + 1,f) = ( *f* f) m"
apply (cases m)
apply (simp add: sumhr star_n_one_num star_n_add starfun)
done
lemma sumhr_add_lbound_zero [simp]: "sumhr(m+k,k,f) = 0"
apply (cases m, cases k)
apply (simp add: sumhr star_n_add star_n_zero_num)
done
lemma sumhr_add: "sumhr (m,n,f) + sumhr(m,n,g) = sumhr(m,n,%i. f i + g i)"
apply (cases m, cases n)
apply (simp add: sumhr star_n_add setsum_addf)
done
lemma sumhr_mult: "hypreal_of_real r * sumhr(m,n,f) = sumhr(m,n,%n. r * f n)"
apply (cases m, cases n)
apply (simp add: sumhr star_of_def star_n_mult setsum_right_distrib)
done
lemma sumhr_split_add: "n < p ==> sumhr(0,n,f) + sumhr(n,p,f) = sumhr(0,p,f)"
apply (cases n, cases p)
apply (auto elim!: FreeUltrafilterNat_subset simp
add: star_n_zero_num sumhr star_n_add star_n_less setsum_add_nat_ivl star_n_eq_iff)
done
lemma sumhr_split_diff: "n<p ==> sumhr(0,p,f) - sumhr(0,n,f) = sumhr(n,p,f)"
by (drule_tac f = f in sumhr_split_add [symmetric], simp)
lemma sumhr_hrabs: "abs(sumhr(m,n,f)) \<le> sumhr(m,n,%i. abs(f i))"
apply (cases n, cases m)
apply (simp add: sumhr star_n_le star_n_abs setsum_abs)
done
text{* other general version also needed *}
lemma sumhr_fun_hypnat_eq:
"(\<forall>r. m \<le> 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)"
by (fastsimp simp add: sumhr hypnat_of_nat_eq intro:setsum_cong)
lemma sumhr_const:
"sumhr(0, n, %i. r) = hypreal_of_hypnat n * hypreal_of_real r"
apply (cases n)
apply (simp add: sumhr star_n_zero_num hypreal_of_hypnat
star_of_def star_n_mult real_of_nat_def)
done
lemma sumhr_less_bounds_zero [simp]: "n < m ==> sumhr(m,n,f) = 0"
apply (cases m, cases n)
apply (auto elim: FreeUltrafilterNat_subset
simp add: sumhr star_n_less star_n_zero_num star_n_eq_iff)
done
lemma sumhr_minus: "sumhr(m, n, %i. - f i) = - sumhr(m, n, f)"
apply (cases m, cases n)
apply (simp add: sumhr star_n_minus setsum_negf)
done
lemma sumhr_shift_bounds:
"sumhr(m+hypnat_of_nat k,n+hypnat_of_nat k,f) = sumhr(m,n,%i. f(i + k))"
apply (cases m, cases n)
apply (simp add: sumhr star_n_add setsum_shift_bounds_nat_ivl hypnat_of_nat_eq)
done
subsection{*Nonstandard Sums*}
text{*Infinite sums are obtained by summing to some infinite hypernatural
(such as @{term whn})*}
lemma sumhr_hypreal_of_hypnat_omega:
"sumhr(0,whn,%i. 1) = hypreal_of_hypnat whn"
by (simp add: hypnat_omega_def star_n_zero_num sumhr hypreal_of_hypnat
real_of_nat_def)
lemma sumhr_hypreal_omega_minus_one: "sumhr(0, whn, %i. 1) = omega - 1"
by (simp add: hypnat_omega_def star_n_zero_num omega_def star_n_one_num
sumhr star_n_diff real_of_nat_def)
lemma sumhr_minus_one_realpow_zero [simp]:
"sumhr(0, whn + whn, %i. (-1) ^ (i+1)) = 0"
by (simp del: realpow_Suc
add: sumhr star_n_add nat_mult_2 [symmetric] star_n_zero_num
star_n_zero_num hypnat_omega_def)
lemma sumhr_interval_const:
"(\<forall>n. m \<le> Suc n --> f n = r) & m \<le> na
==> sumhr(hypnat_of_nat m,hypnat_of_nat na,f) =
(hypreal_of_nat (na - m) * hypreal_of_real r)"
by(simp add: sumhr hypreal_of_nat_eq hypnat_of_nat_eq
real_of_nat_def star_of_def star_n_mult cong: setsum_ivl_cong)
lemma starfunNat_sumr: "( *f* (%n. setsum f {0..<n})) N = sumhr(0,N,f)"
apply (cases N)
apply (simp add: star_n_zero_num starfun sumhr)
done
lemma sumhr_hrabs_approx [simp]: "sumhr(0, M, f) @= sumhr(0, N, f)
==> abs (sumhr(M, N, f)) @= 0"
apply (cut_tac x = M and y = N in linorder_less_linear)
apply (auto simp add: approx_refl)
apply (drule approx_sym [THEN approx_minus_iff [THEN iffD1]])
apply (auto dest: approx_hrabs
simp add: sumhr_split_diff diff_minus [symmetric])
done
(*----------------------------------------------------------------
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"
by (simp add: NSsums_def NSsummable_def, blast)
lemma NSsummable_NSsums: "NSsummable f ==> f NSsums (NSsuminf f)"
apply (simp add: NSsummable_def NSsuminf_def NSsums_def)
apply (blast intro: theI NSLIMSEQ_unique)
done
lemma NSsums_unique: "f NSsums s ==> (s = NSsuminf f)"
by (simp add: suminf_NSsuminf_iff [symmetric] sums_NSsums_iff sums_unique)
lemma NSseries_zero:
"\<forall>m. n \<le> Suc m --> f(m) = 0 ==> f NSsums (setsum f {0..<n})"
by (simp add: sums_NSsums_iff [symmetric] series_zero)
lemma NSsummable_NSCauchy:
"NSsummable f =
(\<forall>M \<in> HNatInfinite. \<forall>N \<in> HNatInfinite. abs (sumhr(M,N,f)) @= 0)"
apply (auto simp add: summable_NSsummable_iff [symmetric]
summable_convergent_sumr_iff convergent_NSconvergent_iff
NSCauchy_NSconvergent_iff [symmetric] NSCauchy_def starfunNat_sumr)
apply (cut_tac x = M and y = N in linorder_less_linear)
apply (auto simp add: approx_refl)
apply (rule approx_minus_iff [THEN iffD2, THEN approx_sym])
apply (rule_tac [2] approx_minus_iff [THEN iffD2])
apply (auto dest: approx_hrabs_zero_cancel
simp add: sumhr_split_diff diff_minus [symmetric])
done
text{*Terms of a convergent series tend to zero*}
lemma NSsummable_NSLIMSEQ_zero: "NSsummable f ==> f ----NS> 0"
apply (auto simp add: NSLIMSEQ_def NSsummable_NSCauchy)
apply (drule bspec, auto)
apply (drule_tac x = "N + 1 " in bspec)
apply (auto intro: HNatInfinite_add_one approx_hrabs_zero_cancel)
done
text{*Nonstandard comparison test*}
lemma NSsummable_comparison_test:
"[| \<exists>N. \<forall>n. N \<le> n --> abs(f n) \<le> g n; NSsummable g |] ==> NSsummable f"
apply (fold summable_NSsummable_iff)
apply (rule summable_comparison_test, simp, assumption)
done
lemma NSsummable_rabs_comparison_test:
"[| \<exists>N. \<forall>n. N \<le> n --> abs(f n) \<le> g n; NSsummable g |]
==> NSsummable (%k. abs (f k))"
apply (rule NSsummable_comparison_test)
apply (auto)
done
end