(* Title : SEQ.thy
Author : Jacques D. Fleuriot
Copyright : 1998 University of Cambridge
Description : Convergence of sequences and series
*)
SEQ = NatStar + HyperPow +
constdefs
(* Standard definition of convergence of sequence *)
LIMSEQ :: [nat=>real,real] => bool ("((_)/ ----> (_))" [60, 60] 60)
"X ----> L == (ALL r. 0 < r --> (EX no. ALL n. no <= n --> abs (X n + -L) < r))"
(* Nonstandard definition of convergence of sequence *)
NSLIMSEQ :: [nat=>real,real] => bool ("((_)/ ----NS> (_))" [60, 60] 60)
"X ----NS> L == (ALL N: HNatInfinite. ( *fNat* X) N @= hypreal_of_real L)"
(* define value of limit using choice operator*)
lim :: (nat => real) => real
"lim X == (@L. (X ----> L))"
nslim :: (nat => real) => real
"nslim X == (@L. (X ----NS> L))"
(* Standard definition of convergence *)
convergent :: (nat => real) => bool
"convergent X == (EX L. (X ----> L))"
(* Nonstandard definition of convergence *)
NSconvergent :: (nat => real) => bool
"NSconvergent X == (EX L. (X ----NS> L))"
(* Standard definition for bounded sequence *)
Bseq :: (nat => real) => bool
"Bseq X == (EX K. (0 < K & (ALL n. abs(X n) <= K)))"
(* Nonstandard definition for bounded sequence *)
NSBseq :: (nat=>real) => bool
"NSBseq X == (ALL N: HNatInfinite. ( *fNat* X) N : HFinite)"
(* Definition for monotonicity *)
monoseq :: (nat=>real)=>bool
"monoseq X == ((ALL (m::nat) n. m <= n --> X m <= X n) |
(ALL m n. m <= n --> X n <= X m))"
(* Definition of subsequence *)
subseq :: (nat => nat) => bool
"subseq f == (ALL m n. m < n --> (f m) < (f n))"
(** Cauchy condition **)
(* Standard definition *)
Cauchy :: (nat => real) => bool
"Cauchy X == (ALL e. (0 < e -->
(EX M. (ALL m n. M <= m & M <= n
--> abs((X m) + -(X n)) < e))))"
NSCauchy :: (nat => real) => bool
"NSCauchy X == (ALL M: HNatInfinite. ALL N: HNatInfinite.
( *fNat* X) M @= ( *fNat* X) N)"
end