The MonadPlus concept represents Monads with a monoidal structure. 
Intuitively, whereas a Monad can be seen as some kind of container or context, a MonadPlus can be seen as a container or a context that can be concatenated with other containers or contexts. There must also be an identity element for this combining operation. For example, a tuple is a MonadPlus, because tuples can be concatenated and the empty tuple would act as an identity for concatenation. How is this different from a Monad which is also a Monoid? The answer is that the monoidal structure on a MonadPlus must not depend of the contents of the structure; it must not require the contents to be a Monoid in order to work.
While sequences are not the only possible model for MonadPlus, the method names used here refer to the MonadPlus of sequences under concatenation. Several useful functions generalizing operations on sequences are included with this concept, like append, prepend and filter.
concat and empty
First, a MonadPlus is required to have a monoidal structure. Hence, it is no surprise that for any MonadPlus M, we require M(T) to be a valid monoid. However, we do not enforce that M(T) actually models the Monoid concept provided by Hana. Further, for all objects a, b, c of data type M(T), 
Secondly, a MonadPlus is also required to obey the following laws, which represent the fact that empty<M(T)>() must be some kind of absorbing element for the chain operation. For all objects a of data type M(T) and functions \( f : T \to M(U) \), 
Functor, Applicative and Monad
| Variables | |
| constexpr auto | boost::hana::append | 
| Append an element to a monadic structure.Given an element xand a monadic structurexs,appendreturns a new monadic structure which is the result of liftingxinto the monadic structure and then combining that (to the right) withxs. In other words,.  More... | |
| constexpr auto | boost::hana::concat | 
| Combine two monadic structures together.Given two monadic structures, concatcombines them together and returns a new monadic structure. The exact definition ofconcatwill depend on the exact model of MonadPlus at hand, but for sequences it corresponds intuitively to simple concatenation.  More... | |
| constexpr auto | boost::hana::cycle | 
| Combine a monadic structure with itself ntimes.Given a monadic structurexsand a non-negative numbern,cyclereturns a new monadic structure which is the result of combiningxswith itselfntimes using theconcatoperation. In other words,.  More... | |
| template<typename M > | |
| constexpr auto | boost::hana::empty | 
| Identity of the monadic combination concat.  More... | |
| constexpr auto | boost::hana::filter | 
| Filter a monadic structure using a custom predicate.Given a monadic structure and a predicate, filterreturns a new monadic structure containing only those elements that satisfy the predicate. This is a generalization of the usualfilterfunction for sequences; it works for any MonadPlus. Intuitively,filteris somewhat equivalent to:  More... | |
| constexpr auto | boost::hana::prefix | 
| Inserts a value before each element of a monadic structure.Given a monadic structure xsand a valuezcalled the prefix,prefixreturns a new monadic structure.prefixsatisfies.  More... | |
| constexpr auto | boost::hana::prepend | 
| Prepend an element to a monadic structure.Given a monadic structure xsand an elementx,prependreturns a new monadic structure which is the result of liftingxinto the monadic structure and then combining that (to the left) withxs. In other words,.  More... | |
| constexpr auto | boost::hana::remove | 
| Remove all the elements of a monadic structure that are equal to some value.Given a monadic structure xsand avalue,removereturns a new monadic structure equal toxswithout all its elements that are equal to the givenvalue.removeis equivalent toremove_ifwith theequal.to(value)predicate, i.e.  More... | |
| constexpr auto | boost::hana::remove_if | 
| Remove all the elements of a monadic structure that satisfy some predicate.Given a monadic structure xsand a unary predicate,remove_ifreturns a new monadic structure equal toxswithout all its elements that satisfy the predicate. This is equivalent tofilterwith a negated predicate, i.e.  More... | |
| template<typename M > | |
| constexpr auto | boost::hana::replicate | 
| Create a monadic structure by combining a lifted value with itself ntimes.Given a valuex, a non-negativeIntegralConstantnand the tag of a monadic structureM,replicatecreates a new monadic structure which is the result of combiningxwith itselfntimes inside the monadic structure. In other words,replicatesimplyliftsxinto the monadic structure, and then combines that with itselfntimes:  More... | |
| constexpr auto | boost::hana::suffix | 
| Inserts a value after each element of a monadic structure.Given a monadic structure xsand a valuez(called the suffix),suffixreturns a new monadic structure such that.  More... | |
| constexpr auto boost::hana::append | 
#include <boost/hana/fwd/append.hpp>
Append an element to a monadic structure.Given an element x and a monadic structure xs, append returns a new monadic structure which is the result of lifting x into the monadic structure and then combining that (to the right) with xs. In other words,. 
 where Xs is the tag of xs. For sequences, this has the intuitive behavior of simply appending an element to the end of the sequence, hence the name.
Rationale for not calling this
push_backSee the rationale for using
prependinstead ofpush_front.
Given a MonadPlus M, the signature is \( \mathtt{append} : M(T) \times T \to M(T) \).
| xs | A monadic structure that will be combined to the left of the element. | 
| x | An element to combine to the right of the monadic structure. | 
| constexpr auto boost::hana::concat | 
#include <boost/hana/fwd/concat.hpp>
Combine two monadic structures together.Given two monadic structures, concat combines them together and returns a new monadic structure. The exact definition of concat will depend on the exact model of MonadPlus at hand, but for sequences it corresponds intuitively to simple concatenation. 
Also note that combination is not required to be commutative. In other words, there is no requirement that
and indeed it does not hold in general.
Given a MonadPlus M, the signature of concat is \( \mathtt{concat} : M(T) \times M(T) \to M(T) \).
| xs,ys | Two monadic structures to combine together. | 
| constexpr auto boost::hana::cycle | 
#include <boost/hana/fwd/cycle.hpp>
Combine a monadic structure with itself n times.Given a monadic structure xs and a non-negative number n, cycle returns a new monadic structure which is the result of combining xs with itself n times using the concat operation. In other words,. 
Also note that since concat is required to be associative, we could also have written 
If n is zero, then the identity of concat, empty, is returned. In the case of sequences, this boils down to returning a sequence containing n copies of itself; for other models it might differ.
Given an IntegralConstant C and a MonadPlus M, the signature is \( \mathrm{cycle} : M(T) \times C \to M(T) \).
| xs | A monadic structure to combine with itself a certain number of times. | 
| n | A non-negative IntegralConstantrepresenting the number of times to combine the monadic structure with itself. Ifnis zero,cyclereturnsempty. | 
| constexpr auto boost::hana::empty | 
#include <boost/hana/fwd/empty.hpp>
Identity of the monadic combination concat. 
Given a MonadPlus M, the signature is \( \mathtt{empty}_M : \emptyset \to M(T) \).
| M | The tag of the monadic structure to return. This must be a model of the MonadPlus concept. | 
| constexpr auto boost::hana::filter | 
#include <boost/hana/fwd/filter.hpp>
Filter a monadic structure using a custom predicate.Given a monadic structure and a predicate, filter returns a new monadic structure containing only those elements that satisfy the predicate. This is a generalization of the usual filter function for sequences; it works for any MonadPlus. Intuitively, filter is somewhat equivalent to: 
 In other words, we basically turn a monadic structure containing [x1, ..., xn] into a monadic structure containing 
 and we then flatten that.
Given a MonadPlus M and an IntegralConstant Bool holding a value of type bool, the signature is \( \mathtt{filter} : M(T) \times (T \to \mathtt{Bool}) \to M(T) \).
| xs | The monadic structure to filter. | 
| pred | A function called as pred(x)for each elementxin the monadic structure and returning whether that element should be kept in the resulting structure. In the current version of the library, the predicate has to return anIntegralConstantholding a value convertible to abool. | 
| constexpr auto boost::hana::prefix | 
#include <boost/hana/fwd/prefix.hpp>
Inserts a value before each element of a monadic structure.Given a monadic structure xs and a value z called the prefix, prefix returns a new monadic structure. prefix satisfies. 
For sequences, this simply corresponds to inserting the prefix before each element of the sequence. For example, given a sequence [x1, ..., xn], prefix will return 
As explained above, this can be generalized to other MonadPlus models, with various levels of interest.
Given a MonadPlus M, the signature is \( \mathrm{prefix} : M(T) \times T \to M(T) \).
| xs | A monadic structure. | 
| pref | A value (the prefix) to insert before each element of a monadic structure. | 
| constexpr auto boost::hana::prepend | 
#include <boost/hana/fwd/prepend.hpp>
Prepend an element to a monadic structure.Given a monadic structure xs and an element x, prepend returns a new monadic structure which is the result of lifting x into the monadic structure and then combining that (to the left) with xs. In other words,. 
For sequences, this has the intuitive behavior of simply prepending an element to the beginning of the sequence, hence the name.
Rationale for not calling this
push_frontWhile
push_frontis the de-facto name used in the standard library, it also strongly suggests mutation of the underlying sequence, which is not the case here. The author also finds thatpush_frontsuggests too strongly the sole interpretation of putting an element to the front of a sequence, whereasprependis slightly more nuanced and bears its name better for e.g.hana::optional.
Given a MonadPlus M, the signature is \( \mathtt{prepend} : M(T) \times T \to M(T) \).
| xs | A monadic structure that will be combined to the right of the element. | 
| x | An element to combine to the left of the monadic structure. | 
| constexpr auto boost::hana::remove | 
#include <boost/hana/fwd/remove.hpp>
Remove all the elements of a monadic structure that are equal to some value.Given a monadic structure xs and a value, remove returns a new monadic structure equal to xs without all its elements that are equal to the given value. remove is equivalent to remove_if with the equal.to(value) predicate, i.e. 
Given a MonadPlus M and a value of type T, the signature is \( \mathrm{remove} : M(T) \times T \to M(T) \)
| xs | A monadic structure to remove some elements from. | 
| value | A value that is compared to every element xof the structure. Elements of the structure that are equal to that value are removed from the structure. This requires every element to be Comparable withvalue. Furthermore, in the current version of the library, comparingvaluewith any element of the structure must yield a compile-time Logical. | 
| constexpr auto boost::hana::remove_if | 
#include <boost/hana/fwd/remove_if.hpp>
Remove all the elements of a monadic structure that satisfy some predicate.Given a monadic structure xs and a unary predicate, remove_if returns a new monadic structure equal to xs without all its elements that satisfy the predicate. This is equivalent to filter with a negated predicate, i.e. 
Given a MonadPlus M and a predicate of type \( T \to Bool \) for some compile-time Logical Bool, the signature is \( \mathrm{remove\_if} : M(T) \times (T \to Bool) \to M(T) \)
| xs | A monadic structure to remove some elements from. | 
| predicate | A unary predicate called as predicate(x), wherexis an element of the structure, and returning whetherxshould be removed from the structure. In the current version of the library,predicatemust return a compile-time Logical. | 
| constexpr auto boost::hana::replicate | 
#include <boost/hana/fwd/replicate.hpp>
Create a monadic structure by combining a lifted value with itself n times.Given a value x, a non-negative IntegralConstant n and the tag of a monadic structure M, replicate creates a new monadic structure which is the result of combining x with itself n times inside the monadic structure. In other words, replicate simply lifts x into the monadic structure, and then combines that with itself n times: 
If n is zero, then the identity of the concat operation is returned. In the case of sequences, this corresponds to creating a new sequence holding n copies of x.
Given an IntegralConstant C and MonadPlus M, the signature is \( \mathtt{replicate}_M : T \times C \to M(T) \).
| M | The tag of the returned monadic structure. It must be a model of the MonadPlus concept. | 
| x | The value to lift into a monadic structure and then combine with itself. | 
| n | A non-negative IntegralConstantrepresenting the number of times to combinelift<M>(x)with itself. Ifn == 0,replicatereturnsempty<M>(). | 
| constexpr auto boost::hana::suffix | 
#include <boost/hana/fwd/suffix.hpp>
Inserts a value after each element of a monadic structure.Given a monadic structure xs and a value z (called the suffix), suffix returns a new monadic structure such that. 
For sequences, this simply corresponds to inserting the suffix after each element of the sequence. For example, given a sequence [x1, ..., xn], suffix will return 
As explained above, this can be generalized to other MonadPlus models, with various levels of interest.
Given a MonadPlus M, the signature is \( \mathtt{suffix} : M(T) \times T \to M(T) \).
| xs | A monadic structure. | 
| sfx | A value (the suffix) to insert after each element of a monadic structure. |