(* 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 HSEQ 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 = (∃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. setsum f {m..<n})) M N" by (simp add: sumhr_def) text{*Base case in definition of @{term sumr}*} lemma sumhr_zero [simp]: "!!m. sumhr (m,0,f) = 0" unfolding sumhr_app by transfer simp text{*Recursive case in definition of @{term 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,%i. f i + g i)" unfolding sumhr_app by transfer (rule setsum_addf [symmetric]) lemma sumhr_mult: "!!m n. hypreal_of_real r * sumhr(m,n,f) = sumhr(m,n,%n. r * f n)" unfolding sumhr_app by transfer (rule setsum_right_distrib) 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: setsum_add_nat_ivl) 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: "!!m n. abs(sumhr(m,n,f)) ≤ sumhr(m,n,%i. abs(f i))" unfolding sumhr_app by transfer (rule setsum_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, %i. r) = hypreal_of_hypnat n * hypreal_of_real r" unfolding sumhr_app by transfer (simp add: real_of_nat_def) 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, %i. - f i) = - sumhr(m, n, f)" unfolding sumhr_app by transfer (rule setsum_negf) lemma sumhr_shift_bounds: "!!m n. sumhr(m+hypnat_of_nat k,n+hypnat_of_nat k,f) = sumhr(m,n,%i. f(i + k))" unfolding sumhr_app by transfer (rule setsum_shift_bounds_nat_ivl) 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: sumhr_const) lemma sumhr_hypreal_omega_minus_one: "sumhr(0, whn, %i. 1) = omega - 1" apply (simp add: sumhr_const) (* FIXME: need lemma: hypreal_of_hypnat whn = omega - 1 *) (* maybe define omega = hypreal_of_hypnat whn + 1 *) apply (unfold star_class_defs omega_def hypnat_omega_def of_hypnat_def star_of_def) apply (simp add: starfun_star_n starfun2_star_n real_of_nat_def) done lemma sumhr_minus_one_realpow_zero [simp]: "!!N. sumhr(0, N + N, %i. (-1) ^ (i+1)) = 0" unfolding sumhr_app by transfer (simp del: realpow_Suc add: nat_mult_2 [symmetric]) 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* (%n. setsum f {0..<n})) N = sumhr(0,N,f)" unfolding sumhr_app by transfer (rule refl) 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: "∀m. n ≤ 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 = (∀M ∈ HNatInfinite. ∀N ∈ 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: "[| ∃N. ∀n. N ≤ n --> abs(f n) ≤ g n; NSsummable g |] ==> NSsummable f" apply (fold summable_NSsummable_iff) apply (rule summable_comparison_test, simp, assumption) done lemma NSsummable_rabs_comparison_test: "[| ∃N. ∀n. N ≤ n --> abs(f n) ≤ g n; NSsummable g |] ==> NSsummable (%k. abs (f k))" apply (rule NSsummable_comparison_test) apply (auto) done end
lemma sumhr_app:
sumhr (M, N, f) = (*f2* (λm n. setsum f {m..<n})) M N
lemma sumhr_zero:
sumhr (m, 0, f) = 0
lemma sumhr_if:
sumhr (m, n + 1, f) = (if n + 1 ≤ m then 0 else sumhr (m, n, f) + (*f* f) n)
lemma sumhr_Suc_zero:
sumhr (n + 1, n, f) = 0
lemma sumhr_eq_bounds:
sumhr (n, n, f) = 0
lemma sumhr_Suc:
sumhr (m, m + 1, f) = (*f* f) m
lemma sumhr_add_lbound_zero:
sumhr (m + k, k, f) = 0
lemma sumhr_add:
sumhr (m, n, f) + sumhr (m, n, g) = sumhr (m, n, λi. f i + g i)
lemma sumhr_mult:
hypreal_of_real r * sumhr (m, n, f) = sumhr (m, n, λn. r * f n)
lemma sumhr_split_add:
n < p ==> sumhr (0, n, f) + sumhr (n, p, f) = sumhr (0, p, f)
lemma sumhr_split_diff:
n < p ==> sumhr (0, p, f) - sumhr (0, n, f) = sumhr (n, p, f)
lemma sumhr_hrabs:
¦sumhr (m, n, f)¦ ≤ sumhr (m, n, λi. ¦f i¦)
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)
lemma sumhr_const:
sumhr (0, n, λi. r) = hypreal_of_hypnat n * hypreal_of_real r
lemma sumhr_less_bounds_zero:
n < m ==> sumhr (m, n, f) = 0
lemma sumhr_minus:
sumhr (m, n, λi. - f i) = - sumhr (m, n, f)
lemma sumhr_shift_bounds:
sumhr (m + hypnat_of_nat k, n + hypnat_of_nat k, f) =
sumhr (m, n, λi. f (i + k))
lemma sumhr_hypreal_of_hypnat_omega:
sumhr (0, whn, λi. 1) = hypreal_of_hypnat whn
lemma sumhr_hypreal_omega_minus_one:
sumhr (0, whn, λi. 1) = ω - 1
lemma sumhr_minus_one_realpow_zero:
sumhr (0, N + N, λi. -1 ^ (i + 1)) = 0
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
lemma starfunNat_sumr:
(*f* (λn. setsum f {0..<n})) N = sumhr (0, N, f)
lemma sumhr_hrabs_approx:
sumhr (0, M, f) ≈ sumhr (0, N, f) ==> ¦sumhr (M, N, f)¦ ≈ 0
lemma sums_NSsums_iff:
f sums l = f NSsums l
lemma summable_NSsummable_iff:
summable f = NSsummable f
lemma suminf_NSsuminf_iff:
suminf f = NSsuminf f
lemma NSsums_NSsummable:
f NSsums l ==> NSsummable f
lemma NSsummable_NSsums:
NSsummable f ==> f NSsums NSsuminf f
lemma NSsums_unique:
f NSsums s ==> s = NSsuminf f
lemma NSseries_zero:
∀m. n ≤ Suc m --> f m = 0 ==> f NSsums setsum f {0..<n}
lemma NSsummable_NSCauchy:
NSsummable f = (∀M∈HNatInfinite. ∀N∈HNatInfinite. ¦sumhr (M, N, f)¦ ≈ 0)
lemma NSsummable_NSLIMSEQ_zero:
NSsummable f ==> f ----NS> 0
lemma NSsummable_comparison_test:
[| ∃N. ∀n≥N. ¦f n¦ ≤ g n; NSsummable g |] ==> NSsummable f
lemma NSsummable_rabs_comparison_test:
[| ∃N. ∀n≥N. ¦f n¦ ≤ g n; NSsummable g |] ==> NSsummable (λk. ¦f k¦)