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-rw-r--r--src/algebra/aggcat.spad.pamphlet24
1 files changed, 23 insertions, 1 deletions
diff --git a/src/algebra/aggcat.spad.pamphlet b/src/algebra/aggcat.spad.pamphlet
index de3c92a9..118ac45a 100644
--- a/src/algebra/aggcat.spad.pamphlet
+++ b/src/algebra/aggcat.spad.pamphlet
@@ -178,6 +178,20 @@ FiniteAggregate(S: Type): Category == Exports where
members: % -> List S
++ \spad{members(u)} returns a list of the consecutive elements of u.
++ For collections, \axiom{members([x,y,...,z]) = (x,y,...,z)}.
+ reduce: ((S,S)->S,%) -> S
+ ++ \spad{reduce(f,u)} reduces the binary operation \spad{f}
+ ++ across \spad{u}. For example, if \spad{u} is \spad{[x,y,...,z]}
+ ++ then \spad{reduce(f,u)} returns \spad{f(..f(f(x,y),...),z)}.
+ ++ Note: if \spad{u} has one element \spad{x},
+ ++ \spad{reduce(f,u)} returns \spad{x}. Error: if \spad{u} is empty.
+ reduce: ((S,S)->S,%,S) -> S
+ ++ \spad{reduce(f,u,x)} reduces the binary operation \spad{f}
+ ++ across \spad{u}, where \spad{x} is the starting value, usually
+ ++ the identity operation of \spad{f}. Same as
+ ++ \spad{reduce(f,u)} if \spad{u} has 2 or more elements.
+ ++ Returns \spad{f(x,y)} if \spad{u} has one element \spad{y},
+ ++ \spad{x} if \spad{u} is empty. For example,
+ ++ \spad{reduce(+,u,0)} returns the sum of the elements of \spad{u}.
if S has BasicType then
count: (S,%) -> NonNegativeInteger
++ \spad{count(x,u)} returns the number of occurrences
@@ -187,6 +201,14 @@ FiniteAggregate(S: Type): Category == Exports where
++ \spad{member?(x,u)} tests if \spad{x} is a member of \spad{u}.
++ For collections,
++ \axiom{member?(x,u) = reduce(or,[x=y for y in u],false)}.
+ reduce: ((S,S)->S,%,S,S) -> S
+ ++ \spad{reduce(f,u,x,z)} reduces the binary operation \spad{f}
+ ++ across \spad{u}, stopping when an "absorbing element"
+ ++ \spad{z} is encountered. As for \spad{reduce(f,u,x)},
+ ++ \spad{x} is the identity operation of \spad{f}.
+ ++ Same as \spad{reduce(f,u,x)} when \spad{u} contains no
+ ++ element \spad{z}. Thus the third argument \spad{x} is
+ ++ returned when u is empty.
add
empty? x == #x = 0
#x == # members x
@@ -196,7 +218,7 @@ FiniteAggregate(S: Type): Category == Exports where
if S has BasicType then
count(s:S, x:%) == count(s = #1, x)
member?(e, x) == any?(e = #1,x)
-
+ reduce(f,x,e,z) == reduce(f,members x,e,z)
@