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SequencesExt.tla
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---------------------------- MODULE SequencesExt ----------------------------
LOCAL INSTANCE Sequences
LOCAL INSTANCE Naturals
LOCAL INSTANCE FiniteSets
LOCAL INSTANCE FiniteSetsExt
LOCAL INSTANCE Functions
LOCAL INSTANCE Folds
(*************************************************************************)
(* Imports the definitions from the modules, but doesn't export them. *)
(*************************************************************************)
-----------------------------------------------------------------------------
ToSet(s) ==
(*************************************************************************)
(* The image of the given sequence s. Cardinality(ToSet(s)) <= Len(s) *)
(* see https://en.wikipedia.org/wiki/Image_(mathematics) *)
(*************************************************************************)
{ s[i] : i \in DOMAIN s }
SetToSeq(S) ==
(**************************************************************************)
(* Convert a set to some sequence that contains all the elements of the *)
(* set exactly once, and contains no other elements. *)
(**************************************************************************)
CHOOSE f \in [1..Cardinality(S) -> S] : IsInjective(f)
TupleOf(set, n) ==
(***************************************************************************)
(* TupleOf(s, 3) = s \X s \X s *)
(***************************************************************************)
[1..n -> set]
SeqOf(set, n) ==
(***************************************************************************)
(* All sequences up to length n with all elements in set. Includes empty *)
(* sequence. *)
(***************************************************************************)
UNION {[1..m -> set] : m \in 0..n}
BoundedSeq(S, n) ==
(***************************************************************************)
(* An alias for SeqOf to make the connection to Sequences!Seq, which is *)
(* the unbounded version of BoundedSeq. *)
(***************************************************************************)
SeqOf(S, n)
-----------------------------------------------------------------------------
Contains(s, e) ==
(**************************************************************************)
(* TRUE iff the element e \in ToSet(s). *)
(**************************************************************************)
\E i \in 1..Len(s) : s[i] = e
Reverse(s) ==
(**************************************************************************)
(* Reverse the given sequence s: Let l be Len(s) (length of s). *)
(* Equals a sequence s.t. << S[l], S[l-1], ..., S[1]>> *)
(**************************************************************************)
[ i \in 1..Len(s) |-> s[(Len(s) - i) + 1] ]
Remove(s, e) ==
(************************************************************************)
(* The sequence s with e removed or s iff e \notin Range(s) *)
(************************************************************************)
SelectSeq(s, LAMBDA t: t # e)
ReplaceAll(s, old, new) ==
(*************************************************************************)
(* Equals the sequence s except that all occurrences of element old are *)
(* replaced with the element new. *)
(*************************************************************************)
LET F[i \in 0..Len(s)] ==
IF i = 0 THEN << >>
ELSE IF s[i] = old THEN Append(F[i-1], new)
ELSE Append(F[i-1], s[i])
IN F[Len(s)]
-----------------------------------------------------------------------------
\* The operators below have been extracted from the TLAPS module
\* SequencesTheorems.tla as of 10/14/2019. The original comments have been
\* partially rewritten.
InsertAt(s, i, e) ==
(**************************************************************************)
(* Inserts element e at the position i moving the original element to i+1 *)
(* and so on. In other words, a sequence t s.t.: *)
(* /\ Len(t) = Len(s) + 1 *)
(* /\ t[i] = e *)
(* /\ \A j \in 1..(i - 1): t[j] = s[j] *)
(* /\ \A k \in (i + 1)..Len(s): t[k + 1] = s[k] *)
(**************************************************************************)
SubSeq(s, 1, i-1) \o <<e>> \o SubSeq(s, i, Len(s))
ReplaceAt(s, i, e) ==
(**************************************************************************)
(* Replaces the element at position i with the element e. *)
(**************************************************************************)
[s EXCEPT ![i] = e]
RemoveAt(s, i) ==
(**************************************************************************)
(* Replaces the element at position i shortening the length of s by one. *)
(**************************************************************************)
SubSeq(s, 1, i-1) \o SubSeq(s, i+1, Len(s))
-----------------------------------------------------------------------------
Cons(elt, seq) ==
(***************************************************************************)
(* Cons prepends an element at the beginning of a sequence. *)
(***************************************************************************)
<<elt>> \o seq
Front(s) ==
(**************************************************************************)
(* The sequence formed by removing its last element. *)
(**************************************************************************)
SubSeq(s, 1, Len(s)-1)
Last(s) ==
(**************************************************************************)
(* The last element of the sequence. *)
(**************************************************************************)
s[Len(s)]
-----------------------------------------------------------------------------
IsPrefix(s, t) ==
(**************************************************************************)
(* TRUE iff the sequence s is a prefix of the sequence t, s.t. *)
(* \E u \in Seq(Range(t)) : t = s \o u. In other words, there exists *)
(* a suffix u that with s prepended equals t. *)
(**************************************************************************)
DOMAIN s \subseteq DOMAIN t /\ \A i \in DOMAIN s: s[i] = t[i]
IsStrictPrefix(s,t) ==
(**************************************************************************)
(* TRUE iff the sequence s is a prefix of the sequence t and s # t *)
(**************************************************************************)
IsPrefix(s, t) /\ s # t
IsSuffix(s, t) ==
(**************************************************************************)
(* TRUE iff the sequence s is a suffix of the sequence t, s.t. *)
(* \E u \in Seq(Range(t)) : t = u \o s. In other words, there exists a *)
(* prefix that with s appended equals t. *)
(**************************************************************************)
IsPrefix(Reverse(s), Reverse(t))
IsStrictSuffix(s, t) ==
(**************************************************************************)
(* TRUE iff the sequence s is a suffix of the sequence t and s # t *)
(**************************************************************************)
IsSuffix(s,t) /\ s # t
-----------------------------------------------------------------------------
SeqMod(a, b) ==
(***************************************************************************)
(* Range(a % b) = 0..b-1, but DOMAIN seq = 1..Len(seq). *)
(* So to do modular arithmetic on sequences we need to *)
(* map 0 to b. *)
(***************************************************************************)
IF a % b = 0 THEN b ELSE a % b
FoldSeq(op(_, _), base, seq) ==
(***************************************************************************)
(* An alias of FoldFunction that op on all elements of seq an arbitrary *)
(* order. The resulting function is: *)
(* op(f[i],op(f[j], ..., op(f[k],base) ...)) *)
(* *)
(* op must be associative and commutative, because we can not assume a *)
(* particular ordering of i, j, and k *)
(* *)
(* Example: *)
(* FoldSeq(LAMBDA x,y: {x} \cup y, {}, <<1,2,1>>) = {1,2} *)
(***************************************************************************)
FoldFunction(op, base, seq)
FoldLeft(op(_, _), base, seq) ==
(***************************************************************************)
(* FoldLeft folds op on all elements of seq from left to right, starting *)
(* with the first element and base. The resulting function is: *)
(* op(op(...op(base,f[0]), ...f[n-1]), f[n]) *)
(* *)
(* *)
(* Example: *)
(* LET cons(x,y) == <<x,y>> *)
(* IN FoldLeft(cons, 0, <<3,1,2>> = << << <<0,3>>, 1>>, 2>> *)
(***************************************************************************)
MapThenFoldSet(LAMBDA x,y : op(y,x), base,
LAMBDA i : seq[i],
LAMBDA s: CHOOSE i \in s : \A j \in s: i >= j,
DOMAIN seq)
FoldRight(op(_, _), seq, base) ==
(***************************************************************************)
(* FoldRight folds op on all elements of seq from right to left, starting *)
(* with the last element and base. The resulting function is: *)
(* op(f[0],op(f[1], ..., op(f[n],base) ...)) *)
(* *)
(* *)
(* Example: *)
(* LET cons(x,y) == <<x,y>> *)
(* IN FoldRight(cons, <<3,1,2>>, 0 ) = << 3, << 1, <<2,0>> >> >> *)
(***************************************************************************)
MapThenFoldSet(op, base,
LAMBDA i : seq[i],
LAMBDA s: CHOOSE i \in s : \A j \in s: i <= j,
DOMAIN seq)
=============================================================================