# Aligned Induction

## Substrate models computation

Haskell commentary on the implementation of Tractable and Practicable Inducers/Substrate models computation

The summed alignment valency-density aligned fud decomposition inducer, $I_{z,\mathrm{Sd,D,F},\infty,\mathrm{n,q}}^{‘}$, is defined above, given non-independent substrate histogram $A \in \mathcal{A}_{z} \setminus \{A^{\mathrm{X}}\}$, as $\begin{eqnarray} &&I_{z,\mathrm{Sd,D,F},\infty,\mathrm{n,q}}^{‘ * }(A) = \\ &&\hspace{2em}\{(D,I_{\approx\mathbf{R}}^{ * }(\mathrm{algnValDensSum}(U_A)(A,D))) : \\ &&\hspace{5em}D \in \mathcal{D}_{\mathrm{F},\infty,U_A,V_A} \cap \mathrm{trees}(\mathcal{S} \times (\mathcal{F}_{\mathrm{n}} \cap \mathcal{F}_{\mathrm{q}})),\\ &&\hspace{8em}\hspace{2em}\forall (C,F) \in \mathrm{cont}(D)~(\mathrm{algn}(A * C * F^{\mathrm{T}})>0)\} \end{eqnarray}$ where $\mathrm{cont}(D) := \mathrm{elements}(\mathrm{contingents}(D))$ and the summed derived alignment valency density, $\mathrm{algnValDensSum}(U) \in \mathcal{A} \times \mathcal{D}_{\mathrm{F}} \to \mathbf{R}$, is defined $\begin{eqnarray} &&\mathrm{algnValDensSum}(U)(A,D) := \\ &&\hspace{5em}\sum (\mathrm{algn}(A * C * F^{\mathrm{T}})/\mathrm{cvl}(F) : (C,F) \in \mathrm{cont}(D)) \end{eqnarray}$ where the derived valency capacity is $\mathrm{cvl}(F) := (w^{1/m} : W = \mathrm{der}(F),~w = |W^{\mathrm{C}}|,~m = |W|)$. The cardinality of the substrate models, $\mathcal{D}_{\mathrm{F},\infty,U_A,V_A} \cap \mathrm{trees}(\mathcal{S} \times (\mathcal{F}_{\mathrm{n}} \cap \mathcal{F}_{\mathrm{q}}))$, has tractable time and space complexities, but may yet be impracticable.

The derived alignment valency-density non-overlapping fud inducer, $I_{z,\mathrm{ad,F,\infty,n,q}}^{‘}$, is defined above as $\begin{eqnarray} &&I_{z,\mathrm{ad,F,\infty,n,q}}^{‘ * }(A) = \\ &&\hspace{5em}\{(F,I_{\approx\mathbf{R}}^{ * }(\mathrm{algn}(A * F^{\mathrm{T}})/\mathrm{cvl}(F))) : F \in \mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{n}} \cap \mathcal{F}_{\mathrm{q}}\} \end{eqnarray}$ The fuds of the decompositions of the fud decomposition inducer, $I_{z,\mathrm{Sd,D,F},\infty,\mathrm{n,q}}^{‘}$, are the substrate fuds of the derived alignment valency-density non-overlapping fud inducer, $I_{z,\mathrm{ad,F,\infty,n,q}}^{‘}$, $\begin{eqnarray} \bigcup \{\mathrm{fuds}(D) : D \in \mathcal{D}_{\mathrm{F},\infty,U_A,V_A} \cap \mathrm{trees}(\mathcal{S} \times (\mathcal{F}_{\mathrm{n}} \cap \mathcal{F}_{\mathrm{q}}))\} &=& \mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{n}} \cap \mathcal{F}_{\mathrm{q}}\end{eqnarray}$ Consider various ways in which the limited-models non-overlapping infinite-layer substrate fuds, $\mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{n}} \cap \mathcal{F}_{\mathrm{q}}$, may be constructed. The infinite-layer substrate fud set $\mathcal{F}_{\infty,U,V} \subset \mathcal{F}_{U,\mathrm{P}}$ is defined as $\mathcal{F}_{\infty,U,V} = \{F : F \subseteq \mathrm{powinf}(U)(V,\emptyset),~\mathrm{und}(F) \subseteq V\}$ where $U$ is the infinite implied system, $U = \mathrm{implied}(\mathrm{filter}(V,U))$, and the infinite power fud $\mathrm{powinf}(U) \in \mathrm{P}(\mathcal{V}_U) \times \mathcal{F}_{U,\mathrm{P}} \to \mathcal{F}_{U,\mathrm{P}}$ is defined without termination $\begin{eqnarray} &&\mathrm{powinf}(U)(V,F) := F \cup G \cup \mathrm{powinf}(U)(V,F \cup G) : \\ &&\hspace{10em}G = \{P^{\mathrm{T}} : K \subseteq \mathrm{vars}(F) \cup V,~P \in \mathrm{B}(K^{\mathrm{CS}})\} \end{eqnarray}$ A tree of non-empty infinite-layer substrate fuds may be constructed such that successive fuds in a path have incremented layer cardinality. That is, the fuds are constructed from bottom-up. Define the infinite partition infinite-layer fud tree $\mathrm{tfi}(U) \in \mathrm{P}(\mathcal{V}_U) \times \mathcal{F}_{U,\mathrm{P}} \to \mathrm{trees}(\mathcal{F}_{U,\mathrm{P}})$ as $\begin{eqnarray} &&\mathrm{tfi}(U)(V,F) := \\ &&\hspace{2em}\{(F \cup G,~\mathrm{tfi}(U)(V,F \cup G)) : \\ &&\hspace{5em}G \subseteq \{P^{\mathrm{T}} : K \subseteq \mathrm{vars}(F) \cup V,~(\mathrm{der}(F) \neq \emptyset \implies K \cap \mathrm{der}(F) \neq \emptyset),\\ &&\hspace{10em}P \in \mathrm{B}(K^{\mathrm{CS}})\},\\ &&\hspace{5em}G \neq \emptyset\} \end{eqnarray}$ Let $\mathrm{tfi}(U) \in \mathrm{P}(\mathcal{V}_U) \to \mathrm{trees}(\mathcal{F}_{U,\mathrm{P}})$ be defined $\mathrm{tfi}(U)(V) = \mathrm{tfi}(U)(V,\emptyset)$.

The fuds of the tree are the infinite-layer substrate fuds $\mathcal{F}_{\infty,U,V} = \mathrm{elements}(\mathrm{tfi}(U)(V)) \cup \{\emptyset\}$ For convenience, define the sets of partition variables in the next layer $\mathrm{tuples} \in \mathrm{P}(\mathcal{V}_U) \times \mathcal{F}_{U,\mathrm{P}^{ * }} \to \mathrm{P}(\mathrm{P}(\mathcal{V}_{U}))$ as (Haskell) $\mathrm{tuples}(V,F) := \{K : K \subseteq \mathrm{vars}(F) \cup V,~(\mathrm{der}(F) \neq \emptyset \implies K \cap \mathrm{der}(F) \neq \emptyset)\}$ The sets of partition variables, $K \in \mathrm{tuples}(V,F)$, will be called tuples in the context of practicable inducers. Note that here a tuple is not an ordered list, although (i) in some implementations tuples have limited cardinalities, and (ii) when ordered they may be used to index an array histogram representation.

The partition infinite-layer fud tree may then be defined more succinctly in terms of tuples as $\begin{eqnarray} &&\mathrm{tfi}(U)(V,F) := \\ &&\hspace{2em}\{(F \cup G,~\mathrm{tfi}(U)(V,F \cup G)) : \\ &&\hspace{5em}G \subseteq \{P^{\mathrm{T}} : K \in \mathrm{tuples}(V,F),~P \in \mathrm{B}(K^{\mathrm{CS}})\},\\ &&\hspace{5em}G \neq \emptyset\} \end{eqnarray}$ The set of next layer fuds, $\{G : G \subseteq \{P^{\mathrm{T}} : K \in \mathrm{tuples}(V,F),~P \in \mathrm{B}(K^{\mathrm{CS}})\},~G \neq \emptyset\} \subset \mathcal{F}_{U,\mathrm{P}}$ may be defined in terms of a set of partition-sets, $\mathrm{P}(\bigcup \{\mathrm{B}(K^{\mathrm{CS}}) : K \in \mathrm{tuples}(V,F)\}) \setminus \{\emptyset\}$. For the first layer, $F = \emptyset$, the set of next layer fuds is the set of fuds of partition transforms of the non-empty partition-sets of the substrate partition-sets set, $\{\{P^{\mathrm{T}} : P \in N\} : N \in \mathcal{N}_{U,V},~N \neq \emptyset\} \subset \mathcal{F}_{U,\mathrm{P}}$ which has cardinality $|\mathcal{N}_{U,V}|-1$. The substrate partition-sets set is defined $\mathcal{N}_{U,V} = \mathrm{P}(\{P : K \subseteq V,~P \in \mathrm{B}(K^{\mathrm{CS}})\})$ For higher layers, $F \neq \emptyset$, the set of next layer fuds corresponds to the intersecting substrate partition-sets set, $\{\{P^{\mathrm{T}} : P \in N\} : N \in \mathcal{N}_{U,W,X}\} \subset \mathcal{F}_{U,\mathrm{P}}$ where $W = \mathrm{vars}(F) \cup V$ and $X = \mathrm{der}(F)$. The intersecting substrate partition-sets set, $\mathcal{N}_{U,V,X}$, is defined, $\mathcal{N}_{U,V,X} = \mathrm{P}(\{P : K \subseteq V,~K \cap X \neq \emptyset,~P \in \mathrm{B}(K^{\mathrm{CS}})\})$ The infinite non-overlapping infinite-layer substrate fuds, $\mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{n}}$, may be similarly constructed by selecting only those fuds which are non-overlapping, $\mathcal{F}_{\infty,U,V} \cap \mathcal{F}_{\mathrm{n}} = \{F : F \in \mathrm{elements}(\mathrm{tfi}(U)(V)),~\neg \mathrm{overlap}(F)\}$

Now consider the construction of the finite limited-models non-overlapping infinite-layer substrate fuds, $\mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{n}} \cap \mathcal{F}_{\mathrm{q}}$, with particular definitions of the limited-models constraints. The limited-models subset of the functional definition sets $\mathcal{F}_{\mathrm{q}} = \mathcal{F}_{\mathrm{u}} \cap \mathcal{F}_{\mathrm{d}} \cap\mathcal{F}_{\mathrm{h}} \cap \mathcal{F}_{\mathrm{b}} \subset \mathcal{F}$ represents the class of subsets of the functional definition sets that are (i) limited-underlying, $\mathcal{F}_{\mathrm{u}}$, (ii) limited-derived, $\mathcal{F}_{\mathrm{d}}$, (iii) limited-layer, $\mathcal{F}_{\mathrm{h}}$ and (iv) limited-breadth, $\mathcal{F}_{\mathrm{b}}$. Here the limited-models constraints are defined explicitly.

In order to be computable, the infinite partition infinite-layer fud tree, $\mathrm{tfi}(U)(V) \in \mathrm{trees}(\mathcal{F}_{U,\mathrm{P}})$, may be made finite by limiting the path length. Define the maximum layer limit as $\mathrm{lmax} \in \mathbf{N}_{>0}$. Define the finite limited-layer partition infinite-layer fud tree $\mathrm{tfih}(U) \in \mathrm{P}(\mathcal{V}_U) \times \mathcal{F}_{U,\mathrm{P}} \times \mathbf{N} \to \mathrm{trees}(\mathcal{F}_{U,\mathrm{P}})$ as $\begin{eqnarray} &&\mathrm{tfih}(U)(V,F,h) := \\ &&\hspace{2em}\{(F \cup G,~\mathrm{tfih}(U)(V,F \cup G,h+1)) : \\ &&\hspace{5em}G \subseteq \{P^{\mathrm{T}} : K \in \mathrm{tuples}(V,F),~P \in \mathrm{B}(K^{\mathrm{CS}})\},\\ &&\hspace{5em}G \neq \emptyset,\\ &&\hspace{5em}h \leq \mathrm{lmax}\} \end{eqnarray}$ Let $\mathrm{tfih}(U) \in \mathrm{P}(\mathcal{V}_U) \to \mathrm{trees}(\mathcal{F}_{U,\mathrm{P}})$ be defined $\mathrm{tfih}(U)(V) = \mathrm{tfih}(U)(V,\emptyset,1)$.

The layer cardinality increments at each step in the tree’s path, so the limited path length constraint, $h \leq \mathrm{lmax}$, could equally well be defined as a limited-layer constraint, $\mathrm{layer}(F \cup G,\mathrm{der}(F \cup G)) \leq \mathrm{lmax}$, although this computation would be longer.

The fuds of the tree are the limited-layer infinite-layer substrate fuds, $\mathcal{F}_{\infty,U_A,V_A} \cap \mathcal{F}_{\mathrm{h}} = \mathrm{elements}(\mathrm{tfih}(U)(V)) \cup \{\emptyset\}$

Define the finite limited-layer limited-underlying-volume limited-breadth partition infinite-layer fud tree $\mathrm{tfiubh}(U) \in \mathrm{P}(\mathcal{V}_U) \times \mathcal{F}_{U,\mathrm{P}} \times \mathbf{N} \to \mathrm{trees}(\mathcal{F}_{U,\mathrm{P}})$ as $\begin{eqnarray} &&\mathrm{tfiubh}(U)(V,F,h) := \\ &&\hspace{2em}\{(F \cup G,~\mathrm{tfiubh}(U)(V,F \cup G,h+1)) : \\ &&\hspace{5em}G \subseteq \{P^{\mathrm{T}} : K \in \mathrm{tuples}(V,F),~|K^{\mathrm{C}}| \leq \mathrm{xmax},~P \in \mathrm{B}(K^{\mathrm{CS}})\},\\ &&\hspace{5em}1 \leq |G| \leq \mathrm{bmax},\\ &&\hspace{5em}h \leq \mathrm{lmax}\} \end{eqnarray}$ where the maximum underlying volume limit is $\mathrm{xmax} \in \mathbf{N}_{>0}$ and the maximum breadth limit is $\mathrm{bmax} \in \mathbf{N}_{>0}$. Let $\mathrm{tfiubh}(U) \in \mathrm{P}(\mathcal{V}_U) \to \mathrm{trees}(\mathcal{F}_{U,\mathrm{P}})$ be defined $\mathrm{tfiubh}(U)(V) = \mathrm{tfiubh}(U)(V,\emptyset,1)$.

The finite set of limited-models non-overlapping infinite-layer substrate fuds is $\begin{eqnarray} \mathcal{F}_{\infty,U,V} \cap \mathcal{F}_{\mathrm{n}} \cap \mathcal{F}_{\mathrm{q}} = \{F : F \in \mathrm{elements}(\mathrm{tfiubh}(U)(V)),~\mathrm{nd}(F)\} \end{eqnarray}$ where $\mathrm{nd} \in \mathcal{F} \to \mathbf{B}$ is defined as $\mathrm{nd}(F) = \neg \mathrm{overlap}(F) \wedge (|W^{\mathrm{C}}| \leq \mathrm{wmax} : W=\mathrm{der}(F))$, and the maximum derived volume limit is $\mathrm{wmax} \in \mathbf{N}_{>0}$.

The set of next layer fuds, $\begin{eqnarray} &&\{G : G \subseteq \{P^{\mathrm{T}} : K \in \mathrm{tuples}(V,F),~|K^{\mathrm{C}}| \leq \mathrm{xmax},~P \in \mathrm{B}(K^{\mathrm{CS}})\},\\ &&\hspace{8em}1 \leq |G| \leq \mathrm{bmax}\} \subset \mathcal{F}_{U,\mathrm{P}} \end{eqnarray}$ may be defined in terms of a set of partition-sets. For the first layer, $F = \emptyset$, the set of next layer fuds is the set of fuds of partition transforms of the non-empty partition-sets of the intersection of the limited-underlying-volume substrate partition-sets set, $\mathcal{N}_{U,V,\mathrm{xmax}}$, and the limited-breadth substrate partition-sets set, $\mathcal{N}_{U,V,\mathrm{bmax}}$, $\{\{P^{\mathrm{T}} : P \in N\} : N \in \mathcal{N}_{U,V,\mathrm{xmax}} \cap \mathcal{N}_{U,V,\mathrm{bmax}},~N \neq \emptyset\} \subset \mathcal{F}_{U,\mathrm{P}}$ which has cardinality $|\mathcal{N}_{U,V,\mathrm{xmax}} \cap \mathcal{N}_{U,V,\mathrm{bmax}}|-1$. The limited-underlying-volume substrate partition-sets set is defined, $\mathcal{N}_{U,V,\mathrm{xmax}} = \mathrm{P}(\{P : K \subseteq V,~|K^{\mathrm{CS}}| \leq \mathrm{xmax},~P \in \mathrm{B}(K^{\mathrm{CS}})\})$ The limited-breadth substrate partition-sets set is defined, $\mathcal{N}_{U,V,\mathrm{bmax}} = \{N : N \in \mathcal{N}_{U,V},~|N| \leq \mathrm{bmax}\}$ For higher layers, $F \neq \emptyset$, the set of next layer fuds corresponds to the intersection of the intersecting substrate partition-sets set, $\mathcal{N}_{U,W,X}$, the limited-underlying-volume substrate partition-sets set, $\mathcal{N}_{U,W,\mathrm{xmax}}$, and the limited-breadth substrate partition-sets set $\mathcal{N}_{U,W,\mathrm{bmax}}$ $\{\{P^{\mathrm{T}} : P \in N\} : N \in \mathcal{N}_{U,W,X} \cap \mathcal{N}_{U,W,\mathrm{xmax}} \cap \mathcal{N}_{U,W,\mathrm{bmax}}\} \subset \mathcal{F}_{U,\mathrm{P}}$ where $W = \mathrm{vars}(F) \cup V$ and $X = \mathrm{der}(F)$.

In the case where the limited-layer limited-underlying-volume limited-breadth partition infinite-layer fud tree, $\mathrm{tfiubh}(U)(V)$, is additionally constrained such that the fuds have derived volume less than or equal to the maximum derived volume limit, $\mathrm{wmax}$, then a subset of the limited-models non-overlapping infinite-layer substrate fuds, $\mathcal{F}_{\infty,U,V} \cap \mathcal{F}_{\mathrm{n}} \cap \mathcal{F}_{\mathrm{q}}$, may be constructed. Define the limited-layer limited-derived-volume limited-underlying-volume limited-breadth partition infinite-layer fud tree $\mathrm{tfiubhd}(U) \in \mathrm{P}(\mathcal{V}_U) \times \mathcal{F}_{U,\mathrm{P}} \times \mathbf{N} \to \mathrm{trees}(\mathcal{F}_{U,\mathrm{P}})$ as $\begin{eqnarray} &&\mathrm{tfiubhd}(U)(V,F,h) := \\ &&\hspace{2em}\{(F \cup G,~\mathrm{tfiubhd}(U)(V,F \cup G,h+1)) : \\ &&\hspace{5em}G \subseteq \{P^{\mathrm{T}} : K \in \mathrm{tuples}(V,F),~|K^{\mathrm{C}}| \leq \mathrm{xmax},~P \in \mathrm{B}(K^{\mathrm{CS}})\},\\ &&\hspace{5em}1 \leq |G| \leq \mathrm{bmax},\\ &&\hspace{5em}W = \mathrm{der}(F \cup G), ~|W^{\mathrm{C}}| \leq \mathrm{wmax},\\ &&\hspace{5em}h \leq \mathrm{lmax}\} \end{eqnarray}$ Again, let $\mathrm{tfiubhd}(U) \in \mathrm{P}(\mathcal{V}_U) \to \mathrm{trees}(\mathcal{F}_{U,\mathrm{P}})$ be defined $\mathrm{tfiubhd}(U)(V) = \mathrm{tfiubhd}(U)(V,\emptyset,1)$.

The limited-derived-volume constraint, $|W^{\mathrm{C}}| \leq \mathrm{wmax}$, is applied at every layer of the fud, so only a subset of the limited-models non-overlapping infinite-layer substrate fuds is searched $\begin{eqnarray} \mathcal{F}_{\infty,U,V} \cap \mathcal{F}_{\mathrm{n}} \cap \mathcal{F}_{\mathrm{q}} \supseteq \{F : F \in \mathrm{elements}(\mathrm{tfiubhd}(U)(V)),~\neg\mathrm{overlap}(F)\} \end{eqnarray}$ For the first layer, $F = \emptyset$, the set of next layer fuds is the set of fuds of partition transforms of the non-empty partition-sets of the intersection of the limited-underlying-volume substrate partition-sets set, $\mathcal{N}_{U,V,\mathrm{xmax}}$, the limited-breadth substrate partition-sets set, $\mathcal{N}_{U,V,\mathrm{bmax}}$, and the limited-derived-volume substrate partition-sets set, $\mathcal{N}_{U,V,\mathrm{wmax}}$, $\{\{P^{\mathrm{T}} : P \in N\} : N \in \mathcal{N}_{U,V,\mathrm{xmax}} \cap \mathcal{N}_{U,V,\mathrm{bmax}} \cap \mathcal{N}_{U,V,\mathrm{wmax}},~N \neq \emptyset\} \subset \mathcal{F}_{U,\mathrm{P}}$ which has cardinality $|\mathcal{N}_{U,V,\mathrm{xmax}} \cap \mathcal{N}_{U,V,\mathrm{bmax}} \cap \mathcal{N}_{U,V,\mathrm{wmax}}|-1$.

The computation of the cardinality of the set of fuds in the higher next layers requires that the set itself be computed, because the limited-derived-volume constraint depends on both the given fud, $F$, and the next layer fud, $G$, for its determination. That is, in some cases the derived variables of the child fud intersect with the derived variables of the given fud, $|\mathrm{der}(F \cup G) \cap \mathrm{der}(F)| \geq 0$. However, the cardinality of the children fuds must be less than or equal to those of the limited-layer limited-underlying-volume limited-breadth partition infinite-layer fud tree, $\mathrm{tfiubh}(U)(V)$, $\begin{eqnarray} &&|\{G : G \subseteq \{P^{\mathrm{T}} : K \in \mathrm{tuples}(V,F),~|K^{\mathrm{C}}| \leq \mathrm{xmax},~P \in \mathrm{B}(K^{\mathrm{CS}})\},\\ &&\hspace{8em}1 \leq |G| \leq \mathrm{bmax},~W = \mathrm{der}(F \cup G), ~|W^{\mathrm{C}}| \leq \mathrm{wmax}\}| \\ &\leq&|\{G : G \subseteq \{P^{\mathrm{T}} : K \in \mathrm{tuples}(V,F),~|K^{\mathrm{C}}| \leq \mathrm{xmax},~P \in \mathrm{B}(K^{\mathrm{CS}})\},\\ &&\hspace{23em}~1 \leq |G| \leq \mathrm{bmax}\}| \end{eqnarray}$

The set of substrate infinite-layer fud decompositions $\mathcal{D}_{\mathrm{F},\infty,U,V}$ is defined such that all of the fuds are infinite-layer substrate fuds and none can appear more than once in a path $\begin{eqnarray} &&\mathcal{D}_{\mathrm{F},\infty,U,V} = \{D : D \in \mathcal{D}_{\mathrm{F,d}},~\mathrm{fuds}(D) \subseteq \mathcal{F}_{\infty,U,V},\\ &&\hspace{2em}\forall L \in \mathrm{paths}(D)~(\mathrm{maxr}(\mathrm{count}(\{(F,i) : (i,(\cdot,F)) \in L\}))=1)\} \end{eqnarray}$ or equivalently, $\begin{eqnarray} \mathcal{D}_{\mathrm{F},\infty,U,V} &=& \{D : D \in \mathcal{D}_{\mathrm{F,d}},~\mathrm{fuds}(D) \subseteq \mathcal{F}_{\infty,U,V},~\forall L \in \mathrm{paths}(D)~(|L| = |\mathrm{ran}(\mathrm{set}(L))|)\} \end{eqnarray}$ The non-empty infinite-layer substrate fud decompositions of non-empty infinite-layer substrate fuds may be constructed by means of a tree of immediate super-decompositions. Define the infinite infinite-layer fud decomposition tree $\mathrm{tdfi}(U) \in \mathrm{P}(\mathcal{V}_U) \times \mathcal{D}_{\mathrm{F,d}} \to \mathrm{trees}(\mathcal{D}_{\mathrm{F,d}})$ as $\begin{eqnarray} &&\mathrm{tdfi}(U)(V,D) := \\ &&\hspace{2em}\{(E,~\mathrm{tdfi}(U)(V,E)) : \\ &&\hspace{5em}Q = \mathrm{paths}(D),~L \in Q,~i \in \{1 \ldots |L|\},\\ &&\hspace{5em}(\cdot,F) = L_{i},~W=\mathrm{der}(F),~S \in W^{\mathrm{CS}},\\ &&\hspace{5em}G \in \mathcal{F}_{\infty,U,V} \setminus (\mathrm{ran}(\mathrm{set}(L_{\{1 \ldots i\}})) \cup \{\emptyset\}),\\ &&\hspace{5em}M=L_{\{1 \ldots i\}} \cup \{(i+1,(S,G))\},\\ &&\hspace{5em}E = \mathrm{tree}(Q \setminus \{L_{\{1 \ldots i\}}\} \cup \{M\}),~E \in \mathcal{D}_{\mathrm{F,d}}\} \end{eqnarray}$ where $\mathrm{tdfi}(U)(V,\emptyset) := \{(E,~\mathrm{tdfi}(U)(V,E)) : G \in \mathcal{F}_{\infty,U,V} \setminus \{\emptyset\},~E =\{((\emptyset,G),\emptyset)\}\}$. Let $\mathrm{tdfi}(U) \in \mathrm{P}(\mathcal{V}_U) \to \mathrm{trees}(\mathcal{D}_{\mathrm{F,d}})$ be defined $\mathrm{tdfi}(U)(V) = \mathrm{tdfi}(U)(V,\emptyset)$.

The infinite infinite-layer fud decomposition tree is constrained to be a tree of immediate super-decompositions, $\begin{eqnarray} &&\mathrm{tdfi}(U)(V,D) = \\ &&\hspace{2em}\{(E,~\mathrm{tdfi}(U)(V,E)) : \\ &&\hspace{5em}E \in \mathcal{D}_{\mathrm{F,d}},~\mathrm{fuds}(E) \subseteq \mathcal{F}_{\infty,U,V} \setminus \{\emptyset\},\\ &&\hspace{5em}\forall L \in \mathrm{paths}(E)~(L_{|L|} \notin \mathrm{set}(L_{\{1 \ldots |L|-1\}})),\\ &&\hspace{5em}D \in \mathrm{subtrees}(E),~|\mathrm{nodes}(E) \setminus \mathrm{nodes}(D)|=1\} \end{eqnarray}$ The decompositions of the tree are a subset of the infinite-layer substrate fud decompositions $\mathcal{D}_{\mathrm{F},\infty,U,V} \supset \mathrm{elements}(\mathrm{tdfi}(U)(V))$ The decompositions of the tree form a proper subset because the empty fud is excluded from the construction.

In this construction exactly one fud is added to the previous decomposition at each step. Thus the cardinality of the fuds equals the position in the path, $\forall L \in \mathrm{paths}(\mathrm{tdfi}(U)(V))~\forall i \in \{1 \ldots |L|\}~(|\mathrm{fuds}(L_i)|=i)$.

A constructed decomposition, $D \in \mathrm{elements}(\mathrm{tdfi}(U)(V))$, may appear more than once in the tree if there are multiple paths to its construction, $|\{M : M \in \mathrm{subpaths}(\mathrm{tdfi}(U)(V)),~M_{|M|} = D\}| \geq 1$, because some decompositions have multiple immediate sub-decompositions.

The limited-models non-overlapping infinite-layer substrate fud decompositions, $\mathcal{D}_{\mathrm{F},\infty,U_A,V_A} \cap \mathrm{trees}(\mathcal{S} \times (\mathcal{F}_{\mathrm{n}} \cap \mathcal{F}_{\mathrm{q}}))$, may also be constructed by the tree of immediate super-decompositions to contain only non-overlapping fuds. Define the finite limited-models non-overlapping infinite-layer fud decomposition tree $\mathrm{tdfinq}(U) \in \mathrm{P}(\mathcal{V}_U) \times \mathcal{D}_{\mathrm{F,d}} \to \mathrm{trees}(\mathcal{D}_{\mathrm{F,d}})$ as $\begin{eqnarray} &&\mathrm{tdfinq}(U)(V,D) := \\ &&\hspace{2em}\{(E,~\mathrm{tdfinq}(U)(V,E)) : \\ &&\hspace{5em}Q = \mathrm{paths}(D),~L \in Q,~i \in \{1 \ldots |L|\},\\ &&\hspace{5em}(\cdot,F) = L_{i},~W=\mathrm{der}(F),~S \in W^{\mathrm{CS}},\\ &&\hspace{5em}G \in \mathcal{F}_{\infty,U,V} \cap \mathcal{F}_{\mathrm{n}} \cap \mathcal{F}_{\mathrm{q}} \setminus (\mathrm{ran}(\mathrm{set}(L_{\{1 \ldots i\}})) \cup \{\emptyset\}),\\ &&\hspace{5em}M=L_{\{1 \ldots i\}} \cup \{(i+1,(S,G))\},\\ &&\hspace{5em}E = \mathrm{tree}(Q \setminus \{L_{\{1 \ldots i\}}\} \cup \{M\}),~E \in \mathcal{D}_{\mathrm{F,d}}\} \end{eqnarray}$ where $\begin{eqnarray} &&\mathrm{tdfinq}(U)(V,\emptyset) := \\ &&\hspace{2em}\{(E,~\mathrm{tdfinq}(U)(V,E)) : \\ &&\hspace{5em}G \in \mathcal{F}_{\infty,U,V} \cap \mathcal{F}_{\mathrm{n}} \cap \mathcal{F}_{\mathrm{q}} \setminus \{\emptyset\},~E = \{((\emptyset,G),\emptyset)\}\} \end{eqnarray}$ Let $\mathrm{tdfinq}(U) \in \mathrm{P}(\mathcal{V}_U) \to \mathrm{trees}(\mathcal{D}_{\mathrm{F,d}})$ be defined $\mathrm{tdfinq}(U)(V) = \mathrm{tdfinq}(U)(V,\emptyset)$.

The decompositions of the tree are a subset the limited-models non-overlapping infinite-layer substrate fud decompositions $\mathcal{D}_{\mathrm{F},\infty,U,V} \cap \mathrm{trees}(\mathcal{S} \times (\mathcal{F}_{\mathrm{n}} \cap \mathcal{F}_{\mathrm{q}})) \supset \mathrm{elements}(\mathrm{tdfinq}(U)(V))$

A construction of a fud decomposition tree can be defined in terms of the finite limited-layer limited-underlying-volume limited-breadth partition infinite-layer fud tree, $\mathrm{tfiubh}(U)(V)$. Define the finite limited-derived non-overlapping limited-layer limited-underlying-volume limited-breadth infinite-layer fud decomposition tree $\mathrm{tdfiubhnd}(U) \in \mathrm{P}(\mathcal{V}_U) \times \mathcal{D}_{\mathrm{F,d}} \to \mathrm{trees}(\mathbf{N}_{>0} \times \mathcal{D}_{\mathrm{F,d}})$ as $\begin{eqnarray} &&\mathrm{tdfiubhnd}(U)(V,D) := \\ &&\hspace{2em}\{((j,E),~\mathrm{tdfiubhnd}(U)(V,E)) : \\ &&\hspace{5em}Q = \mathrm{paths}(D),~L \in Q,~i \in \{1 \ldots |L|\},\\ &&\hspace{5em}(\cdot,F) = L_{i},~W=\mathrm{der}(F),~S \in W^{\mathrm{CS}},\\ &&\hspace{5em}P \in \mathrm{order}(D_{\mathrm{tfiubh}}, \mathrm{subpaths}(\mathrm{tfiubh}(U)(V))),\\ &&\hspace{5em}N = \{(j,M_{|M|}) : (M,j) \in P\},\\ &&\hspace{5em}j \in \{1 \ldots |N|\},~N_j \notin \mathrm{set}(L_{\{1 \ldots i\}}),~\mathrm{nd}(N_j),\\ &&\hspace{5em}M=L_{\{1 \ldots i\}} \cup \{(i+1,(S,N_j))\},\\ &&\hspace{5em}E = \mathrm{tree}(Q \setminus \{L_{\{1 \ldots i\}}\} \cup \{M\}),~E \in \mathcal{D}_{\mathrm{F,d}}\} \end{eqnarray}$ where $\begin{eqnarray} &&\mathrm{tdfiubhnd}(U)(V,\emptyset) := \\ &&\hspace{2em}\{((j,E),~\mathrm{tdfiubhnd}(U)(V,E)) : \\ &&\hspace{5em}P \in \mathrm{order}(D_{\mathrm{tfiubh}}, \mathrm{subpaths}(\mathrm{tfiubh}(U)(V))),\\ &&\hspace{5em}N = \{(j,M_{|M|}) : (M,j) \in P\},\\ &&\hspace{5em}j \in \{1 \ldots |N|\},~\mathrm{nd}(N_j),~E = \{((\emptyset,N_j),\emptyset)\}\} \end{eqnarray}$ Let $\mathrm{tdfiubhnd}(U) \in \mathrm{P}(\mathcal{V}_U) \to \mathrm{trees}(\mathcal{D}_{\mathrm{F,d}})$ be defined $\mathrm{tdfiubhnd}(U)(V) = \mathrm{tdfiubhnd}(U)(V,\emptyset)$.

Here search list $N \in \mathcal{L}(\mathcal{F}_{\infty,U,V} \cap \mathcal{F}_{\mathrm{u}} \cap \mathcal{F}_{\mathrm{b}})$ is constructed given some order $D_{\mathrm{tfiubh}}$ on the subpaths of the finite limited-layer limited-underlying-volume limited-breadth partition infinite-layer fud tree, $\mathrm{tfiubh}(U)(V)$, so that the search enumeration is $P \in \mathrm{enums}(\mathrm{subpaths}(\mathrm{tfiubh}(U)(V)))$ and the finite search list is $N = \{(j,M_{|M|}) : (M,j) \in P\}$.

Again, the decompositions of the tree are a subset of the limited-models non-overlapping infinite-layer substrate fud decompositions $\mathcal{D}_{\mathrm{F},\infty,U,V} \cap \mathrm{trees}(\mathcal{S} \times (\mathcal{F}_{\mathrm{n}} \cap \mathcal{F}_{\mathrm{q}})) \supset \mathrm{ran}(\mathrm{elements}(\mathrm{tdfiubhnd}(U)(V)))$ but the cardinality limited-derived non-overlapping limited-underlying limited-breadth infinite-layer fud decomposition tree is greater than or equal to the cardinality of the limited-models non-overlapping infinite-layer fud decomposition tree, $|\mathrm{tdfiubhnd}(U)(V)| \geq |\mathrm{tdfinq}(U)(V)|$. Therefore a computer $I_{\mathrm{tdfiubhnd}} \in \mathrm{computers}$ that is defined such that its application to the substrate variables, $V$, constructs the limited-models non-overlapping infinite-layer substrate fud decompositions, $I_{\mathrm{tdfiubhnd}}^{ * }(V) \subset \mathcal{D}_{\mathrm{F},\infty,U,V} \cap \mathrm{trees}(\mathcal{S} \times (\mathcal{F}_{\mathrm{n}} \cap \mathcal{F}_{\mathrm{q}}))$, by traversing the limited-derived non-overlapping limited-underlying limited-breadth infinite-layer fud decomposition tree, $\mathrm{tdfiubhnd}(U)(V)$, is such that $I_{\mathrm{tdfiubhnd}}^{\mathrm{t}}(V) > I_{\mathrm{tdfinq}}^{\mathrm{t}}(V)$.

Instead of constructing non-overlapping infinite-layer substrate fuds, $\mathcal{F}_{\infty,U,V} \cap \mathcal{F}_{\mathrm{n}}$, from the partition transforms of tuples, consider constructing them from contracted non-overlapping substrate transforms of tuples. Define the infinite contracted non-overlapping substrate transform infinite-layer fud tree $\mathrm{tfitn}(U) \in \mathrm{P}(\mathcal{V}_U) \times \mathcal{F}_{U,\mathrm{P}^{ * }} \to \mathrm{trees}(\mathcal{F}_{U,\mathrm{P}^{ * }})$ as $\begin{eqnarray} &&\mathrm{tfitn}(U)(V,F) := \\ &&\hspace{2em}\{(F \cup G,~\mathrm{tfitn}(U)(V,F \cup G)) : \\ &&\hspace{5em}G \subseteq \{N^{\mathrm{T}} : K \in \mathrm{tuples}(V,F),~N \in \mathcal{N’}_{U,K,\mathrm{n}} \setminus \{\emptyset\}\},\\ &&\hspace{5em}G \neq \emptyset\} \end{eqnarray}$ Let $\mathrm{tfitn}(U) \in \mathrm{P}(\mathcal{V}_U) \to \mathrm{trees}(\mathcal{F}_{U,\mathrm{P}^{ * }})$ be defined $\mathrm{tfitn}(U)(V) = \mathrm{tfitn}(U)(V,\emptyset)$.

Here the weak non-overlapping substrate partition-sets set, $\mathcal{N’}_{U,K,\mathrm{n}}$, is defined $\mathcal{N’}_{U,K,\mathrm{n}} = \{N : Y \in \mathrm{B’}(K),~N \in \prod_{J \in Y} \mathrm{B}(J^{\mathrm{CS}})\} \cup \{\emptyset\}$ and the non-overlapping substrate transforms set is defined in terms of the weak non-overlapping substrate partition-sets set $\begin{eqnarray} \mathcal{T}_{U,K,\mathrm{n}} = \{N^{\mathrm{T}K} : N \in \mathcal{N’}_{U,K,\mathrm{n}}\} \end{eqnarray}$ The tree is a tree of multi-partition fuds. The partition fuds are a subset of the multi-partition fuds, $\mathcal{F}_{U,\mathrm{P}} \subset \mathcal{F}_{U,\mathrm{P}^{ * }}$, and the non-overlapping substrate transforms are a superset of the partition transforms, $\{P^{\mathrm{T}} : P \in \mathrm{B}(K^{\mathrm{CS}})\} \subseteq \mathcal{T}_{U,K,\mathrm{n}}$, so the infinite non-overlapping infinite-layer substrate fuds can be constructed from the partition fuds in the tree, $\begin{eqnarray} &&\mathcal{F}_{\infty,U,V} \cap \mathcal{F}_{\mathrm{n}} = \\ &&\hspace{2em}\{F : F \in \mathrm{elements}(\mathrm{tfitn}(U)(V)) \cap \mathcal{F}_{U,\mathrm{P}},~\neg \mathrm{overlap}(F)\} \end{eqnarray}$ However, the infinite non-overlapping infinite-layer substrate fuds can also be constructed by exploding the contracted non-overlapping substrate transforms of the multi-partition fuds $\begin{eqnarray} &&\mathcal{F}_{\infty,U,V} \cap \mathcal{F}_{\mathrm{n}} = \\ &&\hspace{2em}\{F’ : F \in \mathrm{elements}(\mathrm{tfitn}(U)(V)),~F’=\mathrm{explode}(F),~\neg \mathrm{overlap}(F’)\} \end{eqnarray}$ where $\mathrm{explode}(F) := \{P^{\mathrm{T}} : (\cdot,W) \in F,~P \in W\} \in \mathcal{F}_{U,\mathrm{P}}$. Note that, although contracted non-overlapping substrate transforms are being added, it is still necessary to explicitly test that the tree exploded fuds are non-overlapping, $\neg \mathrm{overlap}(\mathrm{explode}(F))$. The addition of contracted non-overlapping substrate transforms does not imply that ancestor exploded fuds are non-overlapping.

The construction of the finite limited-models non-overlapping infinite-layer substrate fuds, $\mathcal{F}_{\infty,U,V} \cap \mathcal{F}_{\mathrm{n}} \cap \mathcal{F}_{\mathrm{q}}$, may also be made with contracted non-overlapping substrate transforms rather than partition transforms. Define the finite limited-layer limited-underlying-volume limited-breadth contracted non-overlapping substrate transform infinite-layer fud tree $\mathrm{tfitnubh}(U) \in \mathrm{P}(\mathcal{V}_U) \times \mathcal{F}_{U,\mathrm{P}^{ * }} \times \mathbf{N} \to \mathrm{trees}(\mathcal{F}_{U,\mathrm{P}^{ * }})$ as $\begin{eqnarray} &&\mathrm{tfitnubh}(U)(V,F,h) := \\ &&\hspace{2em}\{(F \cup G,~\mathrm{tfitnubh}(U)(V,F \cup G,h+1)) : \\ &&\hspace{5em}G \subseteq \{N^{\mathrm{T}} : K \in \mathrm{tuples}(V,F),~|K^{\mathrm{C}}| \leq \mathrm{xmax},~N \in \mathcal{N’}_{U,K,\mathrm{n}} \setminus \{\emptyset\}\},\\ &&\hspace{5em}1 \leq |\mathrm{explode}(G)| \leq \mathrm{bmax},\\ &&\hspace{5em}h \leq \mathrm{lmax}\} \end{eqnarray}$ Again, let $\mathrm{tfitnubh}(U) \in \mathrm{P}(\mathcal{V}_U) \to \mathrm{trees}(\mathcal{F}_{U,\mathrm{P}^{ * }})$ be defined $\mathrm{tfitnubh}(U)(V) = \mathrm{tfitnubh}(U)(V,\emptyset,1)$.

The finite set of limited-models non-overlapping infinite-layer substrate fuds is $\begin{eqnarray} &&\mathcal{F}_{\infty,U,V} \cap \mathcal{F}_{\mathrm{n}} \cap \mathcal{F}_{\mathrm{q}} = \\ &&\hspace{2em}\{F’ : M \in \mathrm{subpaths}(\mathrm{tfitnubh}(U)(V)),~F’ =\mathrm{explode}(M_{|M|}),~\mathrm{nd}(F’)\} \end{eqnarray}$ Similarly to the case of the limited-layer limited-underlying-volume limited-breadth partition infinite-layer fud tree, $\mathrm{tfiubh}(U)(V)$, above, a finite computer $I_{\mathrm{tfitnq}} \in \mathrm{computers}$ can be defined such that its application to the substrate variables, $V$, constructs the limited-models non-overlapping infinite-layer substrate fuds, $I_{\mathrm{tfitnq}}^{ * }(V) = \mathcal{F}_{\infty,U,V} \cap \mathcal{F}_{\mathrm{n}} \cap \mathcal{F}_{\mathrm{q}}$, by traversing the limited-layer limited-underlying-volume limited-breadth contracted non-overlapping substrate transform infinite-layer fud tree, $\mathrm{tfitnubh}(U)(V)$, such that all of the paths are searched in sequence. The cardinality of the contracted non-overlapping substrate transform searched list is greater than or equal to the cardinality of the partition transform searched list, $|\mathrm{subpaths}(\mathrm{tfitnubh}(U)(V))| \geq |\mathrm{subpaths}(\mathrm{tfiubh}(U)(V))|$ so the computation time must be greater than or equal to that of the previous case, $I_{\mathrm{tfitnq}}^{\mathrm{t}}(V) \geq I_{\mathrm{tfinq}}^{\mathrm{t}}(V)$.

The limited-layer limited-underlying-volume limited-breadth contracted non-overlapping substrate transform infinite-layer fud tree, $\mathrm{tfitnubh}(U)(V)$, does not readily yield a polynomial-complexity computation of the regular cardinality set of next layer fuds. Consider a restricted variation that limits the tuple derived dimension to a parameter $\mathrm{mmax} \in \mathbf{N}_{>0}$. The tuple derived dimension limit is also constrained such that the breadth limit is a multiple, $\mathrm{bmax}/\mathrm{mmax} \in \mathbf{N}_{>0}$. Define the finite limited-layer limited-tuple-derived-dimension limited-underlying-volume limited-breadth contracted non-overlapping substrate transform infinite-layer fud tree $\mathrm{tfitnmubh}(U) \in \mathrm{P}(\mathcal{V}_U) \times \mathcal{F}_{U,\mathrm{P}^{ * }} \times \mathbf{N} \to \mathrm{trees}(\mathcal{F}_{U,\mathrm{P}^{ * }})$ as $\begin{eqnarray} &&\mathrm{tfitnmubh}(U)(V,F,h) := \\ &&\hspace{2em}\{(F \cup G,~\mathrm{tfitnmubh}(U)(V,F \cup G,h+1)) : \\ &&\hspace{5em}G \subseteq \{N^{\mathrm{T}} : K \in \mathrm{tuples}(V,F),~|K^{\mathrm{C}}| \leq \mathrm{xmax},~N \in \mathcal{N}_{U,K,\mathrm{n},\mathrm{mmax}}\},\\ &&\hspace{5em}1 \leq |G| \leq \mathrm{bmax}/\mathrm{mmax},\\ &&\hspace{5em}h \leq \mathrm{lmax}\} \end{eqnarray}$ Again, let $\mathrm{tfitnmubh}(U) \in \mathrm{P}(\mathcal{V}_U) \to \mathrm{trees}(\mathcal{F}_{U,\mathrm{P}^{ * }})$ be defined $\mathrm{tfitnmubh}(U)(V) = \mathrm{tfitnmubh}(U)(V,\emptyset,1)$.

Here the limited-tuple-derived-dimension non-overlapping substrate partition-sets set, $\mathcal{N}_{U,K,\mathrm{n},\mathrm{mmax}}$, is defined as the limited-breadth non-overlapping substrate partition-sets set, $\mathcal{N}_{U,V,\mathrm{n},\mathrm{bmax}}$, applied to the tuple, $\begin{eqnarray} \mathcal{N}_{U,K,\mathrm{n},\mathrm{mmax}} &=& \{N : Y \in \mathrm{B}(K),~|Y| \leq \mathrm{mmax},~N \in \prod_{J \in Y} \mathrm{B}(J^{\mathrm{CS}})\} \end{eqnarray}$ In the case where $\mathrm{mmax} \leq |K|$, $\begin{eqnarray} \mathcal{N}_{U,K,\mathrm{n},\mathrm{mmax}} = \{N : m \in \{1 \ldots \mathrm{mmax}\},~Y \in \mathrm{S}(K,m),~N \in \prod_{J \in Y} \mathrm{B}(J^{\mathrm{CS}})\} \end{eqnarray}$

The limited-layer limited-tuple-derived-dimension limited-underlying-volume limited-breadth contracted non-overlapping substrate transform infinite-layer fud tree is defined with the limited-tuple-derived-dimension non-overlapping substrate partition-sets set, $\mathcal{N}_{U,K,\mathrm{n},\mathrm{mmax}}$. This is a strong partition-sets set so it excludes the empty transform, $(\emptyset,\emptyset)$, and the unary partition transform, $\{\emptyset^{\mathrm{CS}}\}^{\mathrm{T}}$. The finite set of strong limited-models non-overlapping infinite-layer substrate fuds is $\begin{eqnarray} &&\{F : F \in \mathcal{F}_{\infty,U,V} \cap \mathcal{F}_{\mathrm{n}} \cap \mathcal{F}_{\mathrm{q}},~\{\emptyset^{\mathrm{CS}}\}^{\mathrm{T}} \notin F\} = \\ &&\hspace{2em}\{F’ : M \in \mathrm{subpaths}(\mathrm{tfitnmubh}(U)(V)),~F’ =\mathrm{explode}(M_{|M|}),~\mathrm{nd}(F’)\} \end{eqnarray}$ The cardinality of the limited-layer limited-tuple-derived-dimension limited-underlying-volume limited-breadth contracted non-overlapping substrate transform infinite-layer fud tree searched list is less than or equal to the cardinality of the limited-layer limited-underlying-volume limited-breadth contracted non-overlapping substrate transform infinite-layer fud tree searched list, because of the additional constraint, $|\mathrm{subpaths}(\mathrm{tfitnmubh}(U)(V))| \leq |\mathrm{subpaths}(\mathrm{tfitnubh}(U)(V))|$

In section ‘Substrate structures’ P203 it is shown that the strong non-overlapping substrate transforms set, $\{N^{\mathrm{T}V} : N \in \mathcal{N}_{U,V,\mathrm{n}}\} \subseteq \mathcal{T}_{U,V,\mathrm{n}}$, can be constructed explicitly in terms of linear fuds of a strong non-overlapping substrate self-cartesian transform, $\{N^{\mathrm{T}} : N \in \mathcal{N}_{U,V,\mathrm{c}} \cap \mathcal{N}_{U,V,\mathrm{n}}\}$, followed by a sequence of strong self non-overlapping substrate decremented transforms, $\{N^{\mathrm{T}} : N \in \mathcal{N}_{U,W,-} \cap \mathcal{N}_{U,W,\mathrm{n,s}}\}$. Let the finite set of contracted decrementing linear non-overlapping fuds $\mathcal{F}_{U,\mathrm{n},-,V} \subset \mathcal{F}_{U,\mathrm{P * }}$ be defined as $\begin{eqnarray} &&\mathcal{F}_{U,\mathrm{n},-,V}\\ &&\hspace{2em}=\{\{N^{\mathrm{T}} : (\cdot,N) \in L\} : M \in \mathcal{N}_{U,V,\mathrm{c}} \cap \mathcal{N}_{U,V,\mathrm{n}},\\ &&\hspace{10em}L \in \mathrm{subpaths}(\{(M,\mathrm{tdec}(U)(M))\})\} \\ &&\hspace{2em}=\{\{N^{\mathrm{T}} : (\cdot,N) \in L\} : Y \in \mathrm{B}(V),~M = \{K^{\mathrm{CS}\{\}} : K \in Y\},\\ &&\hspace{10em}L \in \mathrm{subpaths}(\{(M,\mathrm{tdec}(U)(M))\})\} \end{eqnarray}$ where the tree of self non-overlapping substrate decremented partition-sets is defined $\mathrm{tdec}(U) \in \mathrm{P}(\mathcal{V}_U) \to \mathrm{trees}(\mathrm{P}(\mathcal{R}_U))$ as $\mathrm{tdec}(U)(M) := \{(N,\mathrm{tdec}(U)(N)) : N \in \mathcal{N}_{U,M,-} \cap \mathcal{N}_{U,M,\mathrm{n,s}}\}$ and $\mathrm{tdec}(U)(\emptyset) :=\emptyset$. Explicitly this is $\begin{eqnarray} &&\mathrm{tdec}(U)(M) := \{(N,\mathrm{tdec}(U)(N)) :\\ &&\hspace{2em}w \in M,~Q \in \mathrm{decs}(\{w\}^{\mathrm{CS}\{\}}),~N = \{Q\} \cup \{\{u\}^{\mathrm{CS}\{\}} : u \in M,~u \neq w\}\} \end{eqnarray}$ where $\mathrm{decs} = \mathrm{decrements} \in \mathcal{R}_U \to \mathrm{P}(\mathcal{R}_U)$ and $\mathrm{decrements}(P) = \{P \setminus \{C,D\} \cup \{C \cup D\} : C,D \in P,~C \neq D\}$ Then $\{N^{\mathrm{T}V} : N \in \mathcal{N}_{U,V,\mathrm{n}}\} = \{F^{\mathrm{T}V} : F \in \mathcal{F}_{U,\mathrm{n},-,V}\}$. Note that the contracted decrementing linear non-overlapping fuds, $\mathcal{F}_{U,\mathrm{n},-,V}$, are multi-partition fuds, $\mathcal{F}_{U,\mathrm{n},-,V} \subset \mathcal{F}_{U,\mathrm{P}^{ * }}$, so are not necessarily substrate fuds, $\mathcal{F}_{U,V}$, because they do not necessarily consist of partition transforms, $\mathcal{F}_{U,V} \subset \mathcal{F}_{U,\mathrm{P}} \subset \mathcal{F}_{U,\mathrm{P}^{ * }}$. The transforms are already contracted so the corresponding substrate fuds can be constructed $\{\mathrm{explode}(F) : F \in \mathcal{F}_{U,\mathrm{n},-,V}\} \subset \mathcal{F}_{U,V}$. Also, the computation of the contracted decrementing linear non-overlapping fuds would not need to check for flattened partition transforms, even if they were substrate fuds, because partitions in the linear fuds are necessarily distinct from all previous partitions in the sequence.

Instead of constructing non-overlapping infinite-layer substrate fuds, $\mathcal{F}_{\infty,U,V} \cap \mathcal{F}_{\mathrm{n}}$, from either (i) partition transforms, $\mathcal{T}_{U,\mathrm{P}}$, or (ii) non-overlapping substrate transforms, $\mathcal{T}_{U,V,\mathrm{n}}$, consider a construction with contracted decrementing linear non-overlapping fuds, $\mathcal{F}_{U,\mathrm{n},-,V}$. Define the infinite contracted decrementing linear non-overlapping fuds infinite-layer fud tree $\mathrm{tfifdn}(U) \in \mathrm{P}(\mathcal{V}_U) \times \mathcal{F}_{U,\mathrm{P}^{ * }} \to \mathrm{trees}(\mathcal{F}_{U,\mathrm{P}^{ * }})$ as $\begin{eqnarray} &&\mathrm{tfifdn}(U)(V,F) := \\ &&\hspace{2em}\{(F \cup \bigcup Q,~\mathrm{tfifdn}(U)(V,F \cup \bigcup Q)) : \\ &&\hspace{5em}Q \subseteq \{H : K \in \mathrm{tuples}(V,F),~H \in \mathcal{F}_{U,\mathrm{n},-,K}\},\\ &&\hspace{5em}Q \neq \emptyset\} \end{eqnarray}$ Note that, because the contracted decrementing linear non-overlapping fuds sometimes have more than one layer, the layer cardinality of the fuds no longer corresponds to the length of the construction path, $\forall L \in \mathrm{subpaths}(\mathrm{tfifdn}(U)(V))~\forall i \in \{1 \ldots |L|\}~(\mathrm{layer}(L_i,\mathrm{der}(L_i)) \geq i)$ The contracted decrementing linear non-overlapping fuds infinite-layer fud tree, $\mathrm{tfifdn}(U)(V)$, is constrained to contain only strong fud elements, so corresponding only to the strong subset of the non-overlapping substrate fuds $\begin{eqnarray} &&\{F : F \in \mathcal{F}_{\infty,U,V} \cap \mathcal{F}_{\mathrm{n}},~\{\emptyset^{\mathrm{CS}}\}^{\mathrm{T}} \notin F\} = \\ &&\hspace{2em}\{F’ : F \in \mathrm{elements}(\mathrm{tfifdn}(U)(V)),~F’=\mathrm{explode}(F),~\neg \mathrm{overlap}(F’)\} \end{eqnarray}$ Again note that, although contracted decrementing linear non-overlapping fuds are being added, it is still necessary to explicitly test that the tree fuds are non-overlapping, $\neg \mathrm{overlap}(\mathrm{explode}(F))$. The addition of contracted decrementing linear non-overlapping fuds does not imply that ancestor fuds are non-overlapping.

Even though only a strong subset of the non-overlapping infinite-layer substrate fuds is computed, the computation time is greatest when constructed with contracted decrementing linear non-overlapping fuds, rather than partition transforms or contracted non-overlapping substrate transforms, because $|\mathcal{F}_{U,\mathrm{n},-,K}| \geq |\mathcal{T}_{U,K,\mathrm{n}}| \geq |\mathrm{B}(K^{\mathrm{CS}})|$. There are sometimes multiple contracted decrementing linear non-overlapping fuds corresponding to a non-overlapping substrate transform, $\mathrm{maxr}(\{(T,|Q|) : (T,Q) \in \{(F,F^{\mathrm{T}V}) : F \in \mathcal{F}_{U,\mathrm{n},-,K}\}^{-1}\}) \geq 1$, because of multiple linear fud paths to the same non-overlapping substrate transform. The cardinality of possible construction paths may be greater than when constructed with partition transforms, $\begin{eqnarray} &&|\{M : M \in \mathrm{subpaths}(\mathrm{tfifdn}(U)(V)),~M_{|M|} = F\}| \\ &\geq& |\{M : M \in \mathrm{subpaths}(\mathrm{tfi}(U)(V)),~M_{|M|} = \mathrm{explode}(F)\}| \end{eqnarray}$ where $F \in \mathrm{elements}(\mathrm{tfifdn}(U)(V))$.

The construction of a strong subset of the limited-models non-overlapping infinite-layer substrate fuds, $\mathcal{F}_{\infty,U,V} \cap \mathcal{F}_{\mathrm{n}} \cap \mathcal{F}_{\mathrm{q}}$, may also be made with contracted decrementing linear non-overlapping fuds. Define the finite limited-layer limited-tuple-derived-dimension limited-underlying-volume limited-breadth contracted decrementing linear non-overlapping fuds infinite-layer fud tree $\mathrm{tfifdnmubh}(U) \in \mathrm{P}(\mathcal{V}_U) \times \mathcal{F}_{U,\mathrm{P}^{ * }} \times \mathbf{N} \to \mathrm{trees}(\mathcal{F}_{U,\mathrm{P}^{ * }})$ as $\begin{eqnarray} &&\mathrm{tfifdnmubh}(U)(V,F,h) := \\ &&\hspace{2em}\{(F \cup \bigcup Q,~\mathrm{tfifdnmubh}(U)(V,F \cup \bigcup Q,h+1)) : \\ &&\hspace{5em}Q \subseteq \{H : K \in \mathrm{tuples}(V,F),~|K^{\mathrm{C}}| \leq \mathrm{xmax},~H \in \mathcal{F}_{U,\mathrm{n},-,K,\mathrm{mmax}}\},\\ &&\hspace{5em}1 \leq |Q| \leq \mathrm{bmax}/\mathrm{mmax},\\ &&\hspace{5em}h \leq \mathrm{lmax}\} \end{eqnarray}$ Again, let $\mathrm{tfifdnmubh}(U) \in \mathrm{P}(\mathcal{V}_U) \to \mathrm{trees}(\mathcal{F}_{U,\mathrm{P}^{ * }})$ be defined $\mathrm{tfifdnmubh}(U)(V) = \mathrm{tfifdnmubh}(U)(V,\emptyset,1)$.

Here the finite set of limited-tuple-derived-dimension contracted decrementing linear non-overlapping fuds $\mathcal{F}_{U,\mathrm{n},-,K,\mathrm{mmax}}$ is defined as $\begin{eqnarray} &&\mathcal{F}_{U,\mathrm{n},-,K,\mathrm{mmax}}\\ &&\hspace{2em}= \{\{N^{\mathrm{T}} : (\cdot,N) \in L\} : M \in \mathcal{N}_{U,K,\mathrm{c}} \cap \mathcal{N}_{U,K,\mathrm{n,mmax}},\\ &&\hspace{10em}L \in \mathrm{subpaths}(\{(M,\mathrm{tdec}(U)(M))\})\} \\ &&\hspace{2em}= \{\{N^{\mathrm{T}} : (\cdot,N) \in L\} : Y \in \mathrm{B}(K),~|Y| \leq \mathrm{mmax},~M = \{J^{\mathrm{CS}\{\}} : J \in Y\},\\ &&\hspace{10em}L \in \mathrm{subpaths}(\{(M,\mathrm{tdec}(U)(M))\})\} \end{eqnarray}$ The set of limited-tuple-derived-dimension contracted decrementing linear non-overlapping fuds is defined with the limited-tuple-derived-dimension non-overlapping substrate partition-sets set, $\mathcal{N}_{U,K,\mathrm{n},\mathrm{mmax}}$, which is defined as the limited-breadth non-overlapping substrate partition-sets set, $\mathcal{N}_{U,V,\mathrm{n,bmax}}$, applied to the tuple.

The limited-layer limited-tuple-derived-dimension limited-underlying-volume limited-breadth contracted decrementing linear non-overlapping fuds infinite-layer fud tree $\mathrm{tfifdnmubh}(U)(V)$, is defined with the set of strong limited-tuple-derived-dimension contracted decrementing linear non-overlapping fuds, $\mathcal{F}_{U,\mathrm{n},-,K,\mathrm{mmax}}$. These exclude the empty transform, $(\emptyset,\emptyset)$, and the unary partition transform, $\{\emptyset^{\mathrm{CS}}\}^{\mathrm{T}}$. The finite set of strong limited-models non-overlapping infinite-layer substrate fuds is $\begin{eqnarray} &&\{F : F \in \mathcal{F}_{\infty,U,V} \cap \mathcal{F}_{\mathrm{n}} \cap \mathcal{F}_{\mathrm{q}},~\{\emptyset^{\mathrm{CS}}\}^{\mathrm{T}} \notin F\} = \\ &&\hspace{2em}\{F’ : M \in \mathrm{subpaths}(\mathrm{tfifdnmubh}(U)(V)),~F’ =\mathrm{explode}(M_{|M|}),~\mathrm{nd}(F’)\} \end{eqnarray}$ The application of the maximum layer limit is applied to the position in the tree path, $h \leq \mathrm{lmax}$, rather than constraining the layer cardinality of the fuds, because the contracted decrementing linear non-overlapping fuds are purely a means of construction and so could be flattened to a single layer, $\mathrm{layer}(\{H^{\mathrm{T}}\},\mathrm{der}(\{H^{\mathrm{T}}\})) = 1$ where $H \in \mathcal{F}_{U,\mathrm{n},-,K}$. That is, the decrementing notionally takes place within each layer.

Similarly to the case (i) of the limited-layer limited-underlying-volume limited-breadth partition infinite-layer fud tree, $\mathrm{tfiubh}(U)(V)$, and the case (ii) of the limited-layer limited-underlying-volume limited-breadth contracted non-overlapping substrate transform infinite-layer fud tree, $\mathrm{tfitnubh}(U)(V)$, above, a finite computer $I_{\mathrm{tfifdnq}} \in \mathrm{computers}$ can be defined such that its application to the substrate variables, $V$, constructs the strong subset of the limited-models non-overlapping infinite-layer substrate fuds, $I_{\mathrm{tfifdnq}}^{ * }(V) \subseteq \mathcal{F}_{\infty,U,V} \cap \mathcal{F}_{\mathrm{n}} \cap \mathcal{F}_{\mathrm{q}}$, by traversing the finite limited-layer limited-tuple-derived-dimension limited-underlying-volume limited-breadth contracted decrementing linear non-overlapping fuds infinite-layer fud tree, $\mathrm{tfifdnmubh}(U)(V)$, such that all paths are searched in sequence. The cardinality of the contracted decrementing linear non-overlapping fuds searched list is greater than or equal to the cardinality of the contracted non-overlapping substrate transform searched list which in turn is greater than or equal to the cardinality of the partition transform searched list, $\begin{eqnarray} |\mathrm{subpaths}(\mathrm{tfifdnmubh}(U)(V))| &\geq& |\mathrm{subpaths}(\mathrm{tfitnub}(U)(V))| \\ &\geq& |\mathrm{subpaths}(\mathrm{tfiubh}(U)(V))| \end{eqnarray}$ so the computation time must be greater than or equal to that of the previous cases, $I_{\mathrm{tfifdnq}}^{\mathrm{t}}(V) \geq I_{\mathrm{tfitnq}}^{\mathrm{t}}(V) \geq I_{\mathrm{tfinq}}^{\mathrm{t}}(V)$.

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