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)abbrev package MAPHACK1 MappingPackageInternalHacks1
++ Author: S.M.Watt and W.H.Burge
++ Date Created:Jan 87
++ Date Last Updated:Feb 92
++ Basic Operations:
++ Related Domains:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ Examples:
++ References:
++ Description: various Currying operations.
MappingPackageInternalHacks1(A: SetCategory): MPcat == MPdef where
NNI ==> NonNegativeInteger
MPcat == with
iter: ((A -> A), NNI, A) -> A
++\spad{iter(f,n,x)} applies \spad{f n} times to \spad{x}.
recur: ((NNI, A)->A, NNI, A) -> A
++\spad{recur(n,g,x)} is \spad{g(n,g(n-1,..g(1,x)..))}.
MPdef == add
iter(g,n,x) ==
for i in 1..n repeat x := g x -- g(g(..(x)..))
x
recur(g,n,x) ==
for i in 1..n repeat x := g(i,x) -- g(n,g(n-1,..g(1,x)..))
x
)abbrev package MAPHACK2 MappingPackageInternalHacks2
++ Description: various Currying operations.
MappingPackageInternalHacks2(A: SetCategory, C: SetCategory):_
MPcat == MPdef where
NNI ==> NonNegativeInteger
MPcat == with
arg1: (A, C) -> A
++\spad{arg1(a,c)} selects its first argument.
arg2: (A, C) -> C
++\spad{arg2(a,c)} selects its second argument.
MPdef == add
arg1(a, c) == a
arg2(a, c) == c
)abbrev package MAPHACK3 MappingPackageInternalHacks3
++ Description: various Currying operations.
MappingPackageInternalHacks3(A: SetCategory, B: SetCategory, C: SetCategory):_
MPcat == MPdef where
NNI ==> NonNegativeInteger
MPcat == with
comp: (B->C, A->B, A) -> C
++\spad{comp(f,g,x)} is \spad{f(g x)}.
MPdef == add
comp(g,h,x) == g h x
)abbrev package MAPPKG1 MappingPackage1
++ Author: S.M.Watt and W.H.Burge
++ Date Created:Jan 87
++ Date Last Updated: November 12, 2007
++ Basic Operations:
++ Related Domains:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ Examples:
++ References:
++ Description: various Currying operations.
MappingPackage1(A:SetCategory): MPcat == MPdef where
NNI ==> NonNegativeInteger
MPcat == with
nullary: A -> (()->A)
++\spad{nullary A} changes its argument into a
++ nullary function.
coerce: A -> (()->A)
++\spad{coerce A} changes its argument into a
++ nullary function.
fixedPoint: (A->A) -> A
++\spad{fixedPoint f} is the fixed point of function \spad{f}.
++ i.e. such that \spad{fixedPoint f = f(fixedPoint f)}.
fixedPoint: (List A->List A, Integer) -> List A
++\spad{fixedPoint(f,n)} is the fixed point of function
++ \spad{f} which is assumed to transform a list of length
++ \spad{n}.
id: A -> A
++\spad{id x} is \spad{x}.
**: (A->A, NNI) -> (A->A)
++\spad{f**n} is the function which is the n-fold application
++ of \spad{f}.
recur: ((NNI, A)->A) -> ((NNI, A)->A)
++\spad{recur(g)} is the function \spad{h} such that
++ \spad{h(n,x)= g(n,g(n-1,..g(1,x)..))}.
MPdef == add
import MappingPackageInternalHacks1(A)
import %peq: (A,A) -> Boolean from Foreign Builtin
a: A
faa: A -> A
f0a: ()-> A
nullary a == a
coerce a == nullary a
fixedPoint faa ==
g0 := GENSYM()$Lisp
g1 := faa g0
%peq(g0,g1) => error "All points are fixed points"
GEQNSUBSTLIST([g0]$Lisp, [g1]$Lisp, g1)$Lisp
fixedPoint(fll, n) ==
g0 := [(GENSYM()$Lisp):A for i in 1..n]
g1 := fll g0
or/[%peq(e0,e1) for e0 in g0 for e1 in g1] =>
error "All points are fixed points"
GEQNSUBSTLIST(g0, g1, g1)$Lisp
-- Composition and recursion.
id a == a
g**n == iter(g, n, #1)
recur fnaa == recur(fnaa, #1, #2)
)abbrev package MAPPKG2 MappingPackage2
++ Description: various Currying operations.
MappingPackage2(A:SetCategory, C:SetCategory): MPcat == MPdef where
NNI ==> NonNegativeInteger
MPcat == with
const: C -> (A ->C)
++\spad{const c} is a function which produces \spad{c} when
++ applied to its argument.
curry: (A ->C, A) -> (()->C)
++\spad{cu(f,a)} is the function \spad{g}
++ such that \spad{g ()= f a}.
constant: (()->C) -> (A ->C)
++\spad{vu(f)} is the function \spad{g}
++ such that \spad{g a= f ()}.
diag: ((A,A)->C) -> (A->C)
++\spad{diag(f)} is the function \spad{g}
++ such that \spad{g a = f(a,a)}.
MPdef == add
import MappingPackageInternalHacks2(A, C)
a: A
c: C
faa: A -> A
f0c: ()-> C
fac: A -> C
faac: (A,A)->C
const c == arg2(#1, c)
curry(fac, a) == fac a
constant f0c == arg2(#1, f0c())
diag faac == faac(#1, #1)
)abbrev package MAPPKG3 MappingPackage3
++ Description: various Currying operations.
MappingPackage3(A:SetCategory, B:SetCategory, C:SetCategory):_
MPcat == MPdef where
NNI ==> NonNegativeInteger
MPcat == with
curryRight: ((A,B)->C, B) -> (A ->C)
++\spad{curryRight(f,b)} is the function \spad{g} such that
++ \spad{g a = f(a,b)}.
curryLeft: ((A,B)->C, A) -> (B ->C)
++\spad{curryLeft(f,a)} is the function \spad{g}
++ such that \spad{g b = f(a,b)}.
constantRight: (A -> C) -> ((A,B)->C)
++\spad{constantRight(f)} is the function \spad{g}
++ such that \spad{g (a,b)= f a}.
constantLeft: (B -> C) -> ((A,B)->C)
++\spad{constantLeft(f)} is the function \spad{g}
++ such that \spad{g (a,b)= f b}.
twist: ((A,B)->C) -> ((B,A)->C)
++\spad{twist(f)} is the function \spad{g}
++ such that \spad{g (a,b)= f(b,a)}.
*: (B->C, A->B) -> (A->C)
++\spad{f*g} is the function \spad{h}
++ such that \spad{h x= f(g x)}.
MPdef == add
import MappingPackageInternalHacks3(A, B, C)
a: A
b: B
c: C
faa: A -> A
f0c: ()-> C
fac: A -> C
fbc: B -> C
fab: A -> B
fabc: (A,B)->C
faac: (A,A)->C
-- Fix left and right arguments as constants.
curryRight(fabc,b) == fabc(#1,b)
curryLeft(fabc,a) == fabc(a, #1)
-- Add left and right arguments which are ignored.
constantRight fac == fac #1
constantLeft fbc == fbc #2
-- Combinators to rearrange arguments.
twist fabc == fabc(#2, #1)
-- Functional composition
fbc*fab == comp(fbc,fab,#1)
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