diff -r d14c877e14b7 -r b3c35d16affe src/mapping.rs
--- a/src/mapping.rs	Tue Feb 20 12:33:16 2024 -0500
+++ b/src/mapping.rs	Mon Feb 03 19:22:16 2025 -0500
@@ -3,115 +3,248 @@
 */
 
 use std::marker::PhantomData;
-use crate::types::{Float};
-use serde::Serialize;
+use std::borrow::Cow;
+use crate::types::{Num, Float, ClosedMul};
 use crate::loc::Loc;
+pub use crate::instance::{Instance, Decomposition, BasicDecomposition, Space};
+use crate::norms::{Norm, NormExponent};
+use crate::operator_arithmetic::{Weighted, Constant};
 
-/// Trait for application of `Self` as a mathematical function or operator on `X`.
-pub trait Apply<X> {
-    type Output;
+/// A mapping from `Domain` to `Self::Codomain`.
+pub trait Mapping<Domain : Space> {
+    type Codomain : Space;
 
     /// Compute the value of `self` at `x`.
-    fn apply(&self, x : X) -> Self::Output;
-}
+    fn apply<I : Instance<Domain>>(&self, x : I) -> Self::Codomain;
 
-/// This blanket implementation is a workaround helper to Rust trait system limitations.
-///
-/// It is introduced because the trait system does not allow blanket implementations of both
-/// [`Apply<X>`] and [`Apply<&'a X>`]. With this, the latter is implemented automatically for
-/// the reference, which can be sufficient to apply the operation in another blanket implementation.
-impl<'a, T, X> Apply<X> for &'a T where T : Apply<X> {
-    type Output = <T as Apply<X>>::Output;
+    #[inline]
+    /// Form the composition `self ∘ other`
+    fn compose<X : Space, T : Mapping<X, Codomain=Domain>>(self, other : T)
+        -> Composition<Self, T>
+    where
+        Self : Sized
+    {
+        Composition{ outer : self, inner : other, intermediate_norm_exponent : () }
+    }
+
 
     #[inline]
-    fn apply(&self, x : X) -> Self::Output {
-        (*self).apply(x)
+    /// Form the composition `self ∘ other`, assigning a norm to the inermediate space
+    fn compose_with_norm<F, X, T, E>(
+        self, other : T, norm : E
+    )  -> Composition<Self, T, E>
+    where
+        Self : Sized,
+        X : Space,
+        T : Mapping<X, Codomain=Domain>,
+        E : NormExponent,
+        Domain : Norm<F, E>,
+        F : Num
+    {
+        Composition{ outer : self, inner : other, intermediate_norm_exponent : norm }
+    }
+
+    /// Multiply `self` by the scalar `a`.
+    #[inline]
+    fn weigh<C>(self, a : C) -> Weighted<Self, C>
+    where
+        Self : Sized,
+        C : Constant,
+        Self::Codomain : ClosedMul<C::Type>,
+    {
+        Weighted { weight : a, base_fn : self }
     }
 }
 
-/// A mapping from `Domain` to `Codomain`.
-///
-/// This is automatically implemented when the relevant [`Apply`] are implemented.
-pub trait Mapping<Domain> : Apply<Domain, Output=Self::Codomain>
-                            + for<'a> Apply<&'a Domain, Output=Self::Codomain> {
-    type Codomain;
-}
-
-impl<Domain, Codomain, T> Mapping<Domain> for T
-where T : Apply<Domain, Output=Codomain> + for<'a> Apply<&'a Domain, Output=Codomain> {
-    type Codomain = Codomain;
-}
-
-
-/// A helper trait alias for referring to [`Mapping`]s from [`Loc<F, N>`] to `F` a [`Float`].
-pub trait RealMapping<F : Float, const N : usize> : Mapping<Loc<F, N>, Codomain = F> {}
+/// Automatically implemented shorthand for referring to [`Mapping`]s from [`Loc<F, N>`] to `F`.
+pub trait RealMapping<F : Float, const N : usize>
+: Mapping<Loc<F, N>, Codomain = F> {}
 
 impl<F : Float, T, const N : usize> RealMapping<F, N> for T
 where T : Mapping<Loc<F, N>, Codomain = F> {}
 
-
-/// Trait for calculation the differential of `Self` as a mathematical function on `X`.
-pub trait Differentiate<X> {
-    type Output;
+/// A helper trait alias for referring to [`Mapping`]s from [`Loc<F, N>`] to [`Loc<F, M>`].
+pub trait RealVectorField<F : Float, const N : usize, const M : usize>
+: Mapping<Loc<F, N>, Codomain = Loc<F, M>> {}
 
-    /// Compute the differential of `self` at `x`.
-    fn differential(&self, x : X) -> Self::Output;
-}
-
+impl<F : Float, T, const N : usize, const M : usize> RealVectorField<F, N, M> for T
+where T : Mapping<Loc<F, N>, Codomain = Loc<F, M>> {}
 
 /// A differentiable mapping from `Domain` to [`Mapping::Codomain`], with differentials
 /// `Differential`.
 ///
-/// This is automatically implemented when the relevant [`Differentiate`] are implemented.
-pub trait DifferentiableMapping<Domain>
-: Mapping<Domain>
-  + Differentiate<Domain, Output=Self::Differential>
-  + for<'a> Differentiate<&'a Domain, Output=Self::Differential>{
-    type Differential;
+/// This is automatically implemented when [`DifferentiableImpl`] is.
+pub trait DifferentiableMapping<Domain : Space> : Mapping<Domain> {
+    type DerivativeDomain : Space;
+    type Differential<'b> : Mapping<Domain, Codomain=Self::DerivativeDomain> where Self : 'b;
+
+    /// Calculate differential at `x`
+    fn differential<I : Instance<Domain>>(&self, x : I) -> Self::DerivativeDomain;
+
+    /// Form the differential mapping of `self`.
+    fn diff(self) -> Self::Differential<'static>;
+
+    /// Form the differential mapping of `self`.
+    fn diff_ref(&self) -> Self::Differential<'_>;
 }
 
+/// Automatically implemented shorthand for referring to differentiable [`Mapping`]s from
+/// [`Loc<F, N>`] to `F`.
+pub trait DifferentiableRealMapping<F : Float, const N : usize>
+: DifferentiableMapping<Loc<F, N>, Codomain = F, DerivativeDomain = Loc<F, N>> {}
 
-impl<Domain, Differential, T> DifferentiableMapping<Domain> for T
-where T : Mapping<Domain>
-          + Differentiate<Domain, Output=Differential>
-          + for<'a> Differentiate<&'a Domain, Output=Differential> {
-    type Differential = Differential;
+impl<F : Float, T, const N : usize> DifferentiableRealMapping<F, N> for T
+where T : DifferentiableMapping<Loc<F, N>, Codomain = F, DerivativeDomain = Loc<F, N>> {}
+
+/// Helper trait for implementing [`DifferentiableMapping`]
+pub trait DifferentiableImpl<X : Space> : Sized {
+    type Derivative : Space;
+
+    /// Compute the differential of `self` at `x`, consuming the input.
+    fn differential_impl<I : Instance<X>>(&self, x : I) -> Self::Derivative;
 }
 
-/// A sum of [`Mapping`]s.
-#[derive(Serialize, Debug, Clone)]
-pub struct Sum<Domain, M : Mapping<Domain>> {
-    components : Vec<M>,
-    _domain : PhantomData<Domain>,
-}
+impl<T, Domain> DifferentiableMapping<Domain> for T
+where
+    Domain : Space,
+    T : Clone + Mapping<Domain> + DifferentiableImpl<Domain>
+{
+    type DerivativeDomain = T::Derivative;
+    type Differential<'b> = Differential<'b, Domain, Self> where Self : 'b;
+    
+    #[inline]
+    fn differential<I : Instance<Domain>>(&self, x : I) -> Self::DerivativeDomain {
+        self.differential_impl(x)
+    }
 
-impl<Domain, M : Mapping<Domain>> Sum<Domain, M> {
-    /// Construct from an iterator.
-    pub fn new<I : Iterator<Item = M>>(iter : I) -> Self {
-        Sum { components : iter.collect(), _domain : PhantomData }
+    fn diff(self) -> Differential<'static, Domain, Self> {
+        Differential{ g : Cow::Owned(self), _space : PhantomData }
+    }
+
+    fn diff_ref(&self) -> Differential<'_, Domain, Self> {
+        Differential{ g : Cow::Borrowed(self), _space : PhantomData }
     }
 }
 
 
-impl<Domain : Copy, M> Apply<Domain> for Sum<Domain, M>
-where M : Mapping<Domain>,
-      M::Codomain : std::iter::Sum {
-    type Output = M::Codomain;
+/// Container for the differential [`Mapping`] of a [`DifferentiableMapping`].
+pub struct Differential<'a, X, G : Clone> {
+    g : Cow<'a, G>,
+    _space : PhantomData<X>
+}
+
+impl<'a, X, G : Clone> Differential<'a, X, G> {
+    pub fn base_fn(&self) -> &G {
+        &self.g
+    }
+}
+
+impl<'a, X, G> Mapping<X> for Differential<'a, X, G>
+where
+    X : Space,
+    G : Clone + DifferentiableMapping<X>
+{
+    type Codomain = G::DerivativeDomain;
+
+    #[inline]
+    fn apply<I : Instance<X>>(&self, x : I) -> Self::Codomain {
+        (*self.g).differential(x)
+    }
+}
 
-    fn apply(&self, x : Domain) -> Self::Output {
-        self.components.iter().map(|c| c.apply(x)).sum()
+/// Container for flattening [`Loc`]`<F, 1>` codomain of a [`Mapping`] to `F`.
+pub struct FlattenedCodomain<X, F, G> {
+    g : G,
+    _phantoms : PhantomData<(X, F)>
+}
+
+impl<F : Space, X, G> Mapping<X> for FlattenedCodomain<X, F, G>
+where
+    X : Space,
+    G: Mapping<X, Codomain=Loc<F, 1>>
+{
+    type Codomain = F;
+
+    #[inline]
+    fn apply<I : Instance<X>>(&self, x : I) -> Self::Codomain {
+        self.g.apply(x).flatten1d()
+    }
+}
+
+/// An auto-trait for constructing a [`FlattenCodomain`] structure for
+/// flattening the codomain of a [`Mapping`] from [`Loc`]`<F, 1>` to `F`.
+pub trait FlattenCodomain<X : Space, F> : Mapping<X, Codomain=Loc<F, 1>> + Sized {
+    /// Flatten the codomain from [`Loc`]`<F, 1>` to `F`.
+    fn flatten_codomain(self) -> FlattenedCodomain<X, F, Self> {
+        FlattenedCodomain{ g : self, _phantoms : PhantomData }
     }
 }
 
-impl<Domain, M> Differentiate<Domain> for Sum<Domain, M>
-where M : DifferentiableMapping<Domain>,
-      M :: Codomain : std::iter::Sum,
-      M :: Differential : std::iter::Sum,
-      Domain : Copy {
+impl<X : Space, F, G : Sized + Mapping<X, Codomain=Loc<F, 1>>> FlattenCodomain<X, F> for G {}
+
+/// Container for dimensional slicing [`Loc`]`<F, N>` codomain of a [`Mapping`] to `F`.
+pub struct SlicedCodomain<'a, X, F, G : Clone, const N : usize> {
+    g : Cow<'a, G>,
+    slice : usize,
+    _phantoms : PhantomData<(X, F)>
+}
 
-    type Output = M::Differential;
+impl<'a, X, F, G, const N : usize> Mapping<X> for SlicedCodomain<'a, X, F, G, N>
+where
+    X : Space,
+    F : Copy + Space,
+    G : Mapping<X, Codomain=Loc<F, N>> + Clone,
+{
+    type Codomain = F;
 
-    fn differential(&self, x : Domain) -> Self::Output {
-        self.components.iter().map(|c| c.differential(x)).sum()
+    #[inline]
+    fn apply<I : Instance<X>>(&self, x : I) -> Self::Codomain {
+        let tmp : [F; N] = (*self.g).apply(x).into();
+        // Safety: `slice_codomain` below checks the range.
+        unsafe { *tmp.get_unchecked(self.slice) }
     }
 }
+
+/// An auto-trait for constructing a [`FlattenCodomain`] structure for
+/// flattening the codomain of a [`Mapping`] from [`Loc`]`<F, 1>` to `F`.
+pub trait SliceCodomain<X : Space, F : Copy, const N : usize>
+    : Mapping<X, Codomain=Loc<F, N>> + Clone + Sized
+{
+    /// Flatten the codomain from [`Loc`]`<F, 1>` to `F`.
+    fn slice_codomain(self, slice : usize) -> SlicedCodomain<'static, X, F, Self, N> {
+        assert!(slice < N);
+        SlicedCodomain{ g : Cow::Owned(self), slice, _phantoms : PhantomData }
+    }
+
+    /// Flatten the codomain from [`Loc`]`<F, 1>` to `F`.
+    fn slice_codomain_ref(&self, slice : usize) -> SlicedCodomain<'_, X, F, Self, N> {
+        assert!(slice < N);
+        SlicedCodomain{ g : Cow::Borrowed(self), slice, _phantoms : PhantomData }
+    }
+}
+
+impl<X : Space, F : Copy, G : Sized + Mapping<X, Codomain=Loc<F, N>> + Clone, const N : usize>
+SliceCodomain<X, F, N>
+for G {}
+
+
+/// The composition S ∘ T. `E` is for storing a `NormExponent` for the intermediate space.
+pub struct Composition<S, T, E = ()> {
+    pub outer : S,
+    pub inner : T,
+    pub intermediate_norm_exponent : E
+}
+
+impl<S, T, X, E> Mapping<X> for Composition<S, T, E>
+where
+    X : Space,
+    T : Mapping<X>,
+    S : Mapping<T::Codomain>
+{
+    type Codomain = S::Codomain;
+
+    #[inline]
+    fn apply<I : Instance<X>>(&self, x : I) -> Self::Codomain {
+        self.outer.apply(self.inner.apply(x))
+    }
+}