%-----------------------------------------------------------------------------
% Suffixes of sequences of countable length.
%
% Author: Jerry James <[email protected]>
%
% This file and its accompanying proof file are distributed under the CC0 1.0
% Universal license: http://creativecommons.org/publicdomain/zero/1.0/.
%
% Version history:
% 2007 Feb 14: PVS 4.0 version
% 2011 May 6: PVS 5.0 version
% 2013 Jan 14: PVS 6.0 version
%-----------------------------------------------------------------------------
csequence_suffix[T: TYPE]: THEORY
BEGIN
IMPORTING csequence_concatenate[T], orders[csequence]
p: VAR pred[T]
n, m: VAR nat
cseq, cseq1, cseq2: VAR csequence
fseq, fseq1, fseq2: VAR finite_csequence
iseq: VAR infinite_csequence
nseq: VAR nonempty_csequence
eseq: VAR empty_csequence
suffix?(cseq1, cseq2): INDUCTIVE bool =
cseq1 = cseq2 OR (nonempty?(cseq2) AND suffix?(cseq1, rest(cseq2)))
suffix?(cseq2)(cseq1): MACRO bool = suffix?(cseq1, cseq2)
suffix?_empty: THEOREM
FORALL eseq, cseq: suffix?(eseq, cseq) IFF is_finite(cseq)
suffix?_rest_left: THEOREM
FORALL nseq, cseq: suffix?(nseq, cseq) IMPLIES suffix?(rest(nseq), cseq)
suffix?_rest_right: THEOREM
FORALL cseq, nseq: suffix?(cseq, rest(nseq)) IMPLIES suffix?(cseq, nseq)
suffix?_finite_left: THEOREM
FORALL fseq, cseq: suffix?(fseq, cseq) IMPLIES is_finite(cseq)
suffix?_finite_right: THEOREM
FORALL cseq, fseq: suffix?(cseq, fseq) IMPLIES is_finite(cseq)
suffix?_infinite_left: THEOREM
FORALL iseq, cseq: suffix?(iseq, cseq) IMPLIES is_infinite(cseq)
suffix?_infinite_right: THEOREM
FORALL cseq, iseq: suffix?(cseq, iseq) IMPLIES is_infinite(cseq)
suffix?_length: THEOREM
FORALL fseq1, fseq2:
suffix?(fseq1, fseq2) IMPLIES length(fseq1) <= length(fseq2)
suffix?_length_eq: THEOREM
FORALL fseq1, fseq2:
suffix?(fseq1, fseq2) AND length(fseq1) = length(fseq2) IMPLIES
fseq1 = fseq2
suffix?_index: THEOREM
FORALL cseq1, cseq2, n:
suffix?(cseq1, cseq2) AND index?(cseq1)(n) IMPLIES index?(cseq2)(n)
suffix?_concatenate: THEOREM FORALL fseq, cseq: suffix?(cseq, fseq o cseq)
suffix?_def: THEOREM
FORALL cseq1, cseq2:
suffix?(cseq1, cseq2) IFF (EXISTS fseq: fseq o cseq1 = cseq2)
% Note that suffix? is not a partial order, because it is not
% antisymmetric. These two sequences are suffixes of each other,
% but are not equal:
%
% 1, 2, 1, 2, 1, 2, ...
% 2, 1, 2, 1, 2, 1, ...
suffix?_is_preorder: JUDGEMENT suffix? HAS_TYPE (preorder?[csequence])
% However, suffix? is a partial order when restricted to finite
% sequences.
suffix?_finite_antisymmetric: THEOREM
partial_order?[finite_csequence]
(restrict
[[csequence, csequence], [finite_csequence, finite_csequence],
bool]
(suffix?))
% A dichotomous?-like property for suffixes of a given sequence.
suffix?_order: THEOREM
FORALL cseq, cseq1, cseq2:
suffix?(cseq1, cseq) AND suffix?(cseq2, cseq) IMPLIES
suffix?(cseq1, cseq2) OR suffix?(cseq2, cseq1)
% The suffix of cseq after skipping over n elements
suffix(cseq, n): RECURSIVE (suffix?(cseq)) =
IF n = 0 OR empty?(cseq) THEN cseq ELSE suffix(rest(cseq), n - 1) ENDIF
MEASURE n
suffix_is_finite: JUDGEMENT suffix(fseq, n) HAS_TYPE finite_csequence
suffix_is_infinite: JUDGEMENT suffix(iseq, n) HAS_TYPE infinite_csequence
suffix_0: THEOREM FORALL cseq: suffix(cseq, 0) = cseq
suffix_1: THEOREM FORALL nseq: suffix(nseq, 1) = rest(nseq)
suffix_rest1: THEOREM
FORALL nseq, n: suffix(rest(nseq), n) = suffix(nseq, n + 1)
suffix_rest2: THEOREM
FORALL cseq, (n: indexes(cseq)):
rest(suffix(cseq, n)) = suffix(cseq, n + 1)
suffix_suffix: THEOREM
FORALL cseq, n, m: suffix(suffix(cseq, n), m) = suffix(cseq, n + m)
suffix_length: THEOREM
FORALL fseq, n: length(suffix(fseq, n)) = max(0, length(fseq) - n)
suffix_first: THEOREM
FORALL nseq, (n: indexes(nseq)): first(suffix(nseq, n)) = nth(nseq, n)
suffix_index: THEOREM
FORALL cseq, n, m: index?(suffix(cseq, n))(m) IFF index?(cseq)(n + m)
suffix_nth: THEOREM
FORALL cseq, n, (m: indexes(suffix(cseq, n))):
nth(suffix(cseq, n), m) = nth(cseq, n + m)
suffix_empty: THEOREM
FORALL cseq, n: empty?(suffix(cseq, n)) IFF NOT index?(cseq)(n)
suffix_nonempty: THEOREM
FORALL cseq, n: nonempty?(suffix(cseq, n)) IFF index?(cseq)(n)
suffix_concatenate: THEOREM
FORALL cseq1, cseq2, n:
suffix(cseq1 o cseq2, n) =
IF index?(cseq1)(n) THEN suffix(cseq1, n) o cseq2
ELSE suffix(cseq2, n - length(cseq1))
ENDIF
suffix?_suffix: THEOREM
FORALL cseq1, cseq2:
suffix?(cseq1, cseq2) IFF (EXISTS n: cseq1 = suffix(cseq2, n))
suffix_some: THEOREM
FORALL cseq, n, p:
some(p)(suffix(cseq, n)) IFF
(EXISTS (i: indexes(cseq)): i >= n AND p(nth(cseq, i)))
suffix_every: THEOREM
FORALL cseq, n, p:
every(p)(suffix(cseq, n)) IFF
(FORALL (i: indexes(cseq)): i >= n IMPLIES p(nth(cseq, i)))
END csequence_suffix
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