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sigma_countable[T:TYPE]: THEORY
%------------------------------------------------------------------------------
% The sigma_set theory introduces and defines properties of the sigma
% function that sums an arbitrary function F: [T -> real] over a countable
% set of values X. To make this sensible for infinite sets, we will also
% require that f is absolutely convergent in the infinite case.
%
% This notation is a vital simplification for discrete probability theory.
%
% ----
% sigma(X, f) = \ f(x)
% /
% ----
% member(x,X)
%
%
% MODIFICATIONS:
%
% Author: David Lester, Manchester University 20/4/05
%
% Additions 20/1/10 DR Lester
%
%------------------------------------------------------------------------------
BEGIN
IMPORTING sets_aux@countable_props[T],
countable_convergence[T]
a: VAR real
X,Y: VAR countable_set
t: VAR T
f,g: VAR [T -> real]
F: VAR non_empty_finite_set[T]
n: VAR nat
phi: VAR (bijective?[nat,nat])
npx: VAR npreal
nnx: VAR nnreal
sigma(X:countable_set[T],f:(convergent?(X))):real
= IF empty?(X)
THEN 0
ELSIF is_finite(X)
THEN sigma[below[card(X)]](0,card(X)-1,f o finite_enumeration(X))
ELSE limit(series(f o denumerable_enumeration(X))) ENDIF
sigma_empty: LEMMA sigma(emptyset[T],g) = 0
sigma_finite: LEMMA LET n = card(F)-1 IN
sigma(F,g) = sigma[below[n+1]](0,n,g o finite_enumeration(F))
sigma_singleton: LEMMA sigma(singleton(t),g) = g(t)
sigma_infinite: LEMMA FORALL (X:countably_infinite_set,f:(convergent?(X))):
sigma(X,f) = limit(series(f o denumerable_enumeration(X)))
nonempty_countable?(X:set[T]):bool = nonempty?(X) AND is_countable(X)
nonempty_countable: TYPE = (nonempty_countable?)
nonempty_countable_is_countable:
JUDGEMENT nonempty_countable SUBTYPE_OF countable_set[T]
nonempty_countable_is_nonempty:
JUDGEMENT nonempty_countable SUBTYPE_OF (nonempty?[T])
sigma_disjoint_union: LEMMA FORALL (f:(convergent?(union(X,Y)))):
disjoint?(X,Y) => sigma(union(X,Y),f) = sigma(X,f) + sigma(Y,f)
sigma_union: LEMMA FORALL (f:(convergent?(union(X,Y)))):
sigma(union(X,Y),f) = sigma(X,f) + sigma(Y,f) - sigma(intersection(X,Y),f)
sigma_intersection: LEMMA FORALL (f:(convergent?(union(X,Y)))):
sigma(intersection(X,Y),f) = sigma(X,f) + sigma(Y,f) - sigma(union(X,Y),f)
sigma_difference: LEMMA FORALL (f:(convergent?(union(X,Y)))):
sigma(difference(X,Y),f)
= sigma(X,f) - sigma(Y,f) + sigma(difference(Y,X),f)
sigma_subset: LEMMA FORALL (f:(convergent?(Y))):
subset?(X,Y) => sigma(Y,f) = sigma(X,f) + sigma(difference(Y,X),f)
sigma_add: LEMMA FORALL (f:(convergent?(X))):
sigma(add(t,X),f) = IF member(t,X) THEN sigma(X,f)
ELSE f(t) + sigma(X,f) ENDIF
sigma_remove: LEMMA FORALL (f:(convergent?(X))):
sigma(remove(t,X),f) = IF member(t,X) THEN sigma(X,f) - f(t)
ELSE sigma(X,f) ENDIF
sigma_choose_rest: LEMMA FORALL (X:nonempty_countable,f:(convergent?(X))):
sigma(X,f) = f(choose(X)) + sigma(rest(X),f)
sigma_zero: LEMMA sigma(X,LAMBDA t: 0) = 0
sigma_scal: LEMMA FORALL (f:(convergent?(X))): sigma(X,a*f) = a*sigma(X,f)
sigma_opp: LEMMA FORALL (f:(convergent?(X))): sigma(X,-f) = -sigma(X,f)
sigma_plus: LEMMA FORALL (f,g:(convergent?(X))):
sigma(X,f+g) = sigma(X,f) + sigma(X,g)
sigma_diff: LEMMA FORALL (f,g:(convergent?(X))):
sigma(X,f-g) = sigma(X,f) - sigma(X,g)
sigma_ge_0: LEMMA FORALL (f:(convergent?(X))):
(FORALL (t:(X)): f(t) >= 0) => sigma(X,f) >= 0
sigma_gt_0: LEMMA FORALL (X:nonempty_countable,f:(convergent?(X))):
(FORALL (t:(X)): f(t) > 0) => sigma(X,f) > 0
sigma_abs: LEMMA FORALL (f:(convergent?(X))):
abs(sigma(X,f)) <= sigma(X,abs(f))
sigma_eq: LEMMA FORALL (f,g:(convergent?(X))):
(FORALL (t:(X)): f(t) = g(t)) => sigma(X,f) = sigma(X,g)
sigma_le: LEMMA FORALL (f,g:(convergent?(X))):
(FORALL (t:(X)): f(t) <= g(t)) => sigma(X,f) <= sigma(X,g)
sigma_lt: LEMMA FORALL (X:nonempty_countable,f,g:(convergent?(X))):
(FORALL (t:(X)): f(t) < g(t)) => sigma(X,f) < sigma(X,g)
IMPORTING reals@bounded_reals,
sigma_bijection_nat
sigma_bij: LEMMA FORALL (f:(convergent?(X)),phi:(bijective?[(X),(X)])):
sigma(X,extend[T,(X),real,0](f o phi)) = sigma(X,f)
sigma_nn_def: LEMMA
FORALL (f:(convergent?(X))):
(FORALL t: X(t) => f(t) >= 0) =>
sigma(X,f)
= sup({nnx | EXISTS (F:finite_set[T]):
subset?(F,X) AND nnx = sigma(F,f)})
sigma_def: LEMMA
FORALL (f:(convergent?(X))):
sigma(X,f)
= sup({nnx | EXISTS (F:finite_set[T]): subset?(F,X) AND
(FORALL t: F(t) => f(t) >= 0) AND
nnx = sigma(F,f)}) +
inf({npx | EXISTS (F:finite_set[T]): subset?(F,X) AND
(FORALL t: F(t) => f(t) <= 0) AND
npx = sigma(F,f)})
END sigma_countable
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