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