core-language.pdc

## Basics

Juvix Core is the core language in the Juvix compiler stack, defining the canonical syntax & semantics on which all compilers & interpreters must agree. Lower-level evaluation choices may safely differ between implementations as long as they respect the core semantics.

Juvix Core is primarily inspired by Quantitative Type Theory [@quantitative-type-theory], Formality [@formality], and Cedille [@the-calculus-of-dependent-lambda-eliminations]. It fuses full-spectrum dependent types (types & terms can depend on types & terms) with linear logic using the contemplation — computation distinction introduced by QTT, adds the self-types of Cedille & Formality to enable derivation of induction for pure lambda terms, introduces the full connective set of linear logic, dependent where appropriate, to express different varieties of conjunction & disjunction, and defines an extension system for opaque user-defined datatypes & primitives (such as integers with addition & multiplication, bytestrings with concatenation & indexing, or cryptographic keys with construction & signature checking).

> Note: at present, the substructural typing in the core language is not required for optimal reduction — separate elementary affine logic assignments are inferred in the lower-level stage. Substructural typing is used in Juvix Core to provide additional precision to the programmer and enable optimizations in the bespoke compilation path to custom rewrite rules (such as avoiding garbage collection). In the future a closer fusion allowing the more precise usage information to inform interaction net construction is expected; this is an open research question.

## Preliminaries

A *semiring* $R$ is a set $R$ with binary operations $+$ (addition) and $⋅$ (multiplication), such that $(R, +)$ is a commutative monoid with identity $0$,
$(R, ⋅)$ is a monoid with identity $1$, multiplication left and right distribute over addition, and multiplication by $0$ annihilates $R$.

The core type theory must be instantiated over a particular semiring. Choices include the boolean semiring $(0, 1)$, the zero-one-many semiring $(0, 1, ω)$, and the natural numbers with addition and multiplication.

In canonical Juvix Core the type theory is instantiated over the semiring of natural numbers plus ω, which is the most expressive option — terms can be $0$-usage ("contemplated"), $n$-usage ("computed $n$ times"), or $ω$-usage ("computed any number of times"). \textbf{<- is $ω$ really necessary? What are the use cases for this? We want to add safety via our language and be as precise and descriptive as possible I think. And what do we gain from natural numbers instead of $(0, 1, ω)$? Maybe the ability to specify $n$-usage instead of many?}

Let $S$ be a set of sorts $(i, j, k)$ with a total order. The typing rule is:

\begin{prooftree}
    \AxiomC{}
    \RightLabel{SORT j > j}
    \UnaryInfC{$Γ ⊢_0 *_j \ni *_i $}
    \end{prooftree}

Let $K$ be the set of primitive types, $C$ be the set of primitive constants, and $⋮$ be the typing relation between primitive constants and primitive types, which must assign to each primitive constant a unique primitive type.

Let $F$ be the set of primitive functions, where each $f_i$ is related to a function type by the $⋮$ relation and endowed with a reduction operation $→_{f_i}$, which provided an argument of the function input type computes an argument of the function output type.

Primitive types, primitive constants, and primitive functions are threaded-through to the untyped lambda calculus to which Juvix Core is erased, so they must be directly supported by the low-level execution model.

## Syntax

Inspired by the bidirectional syntax of Conor McBride in I Got Plenty o’ Nuttin’ [@plenty-o-nuttin], let $R, S, T, s, t$ be types & terms and $d, e, f$ be eliminations, where types can be synthesized for eliminations but must be specified in advance for terms.

\begin{figure}[H]
\caption{Core syntax}
\begin{subfigure}[t]{0.7\textwidth}
\begin{align*}
R, S, T, s, t &::= ∗_i\ & \text{sort $i$} &\ \text{Star} \\
&\ \ \ \ \ \ |\ \kappa \in K & \text{primitive type} &\ \text{k} \\
&\ \ \ \ \ \ |\ (x \overset{π}{:} S) → T\ & \text{function type} &\ \text{Pi} \\
&\ \ \ \ \ \ |\ (x \overset{π}{:} S) ⊗ T\ & \text{dependent multiplicative conjunction type} &\ (x) \\
&\ \ \ \ \ \ |\ (x \overset{π}{:} S)\ \&\ T\ & \text{dependent additive conjunction type} &\ / \textbackslash \\
&\ \ \ \ \ \ |\ T \parr T & \text{non-dependent multiplicative disjunction type} &\ \textbackslash /  \\
&\ \ \ \ \ \ |\ ιx.T\ & \text{self-type} &\ \\
&\ \ \ \ \ \ |\ λx.t\ & \text{abstraction} &\ \\
&\ \ \ \ \ \ |\ e\ & \text{elimination} &\ \\
\end{align*}
\end{subfigure}

\begin{subfigure}[t]{0.7\textwidth}
\begin{align*}
d, e, f &::= x\ & \text{variable} &\ \\
&\ \ \ \ \ \ |\ c \in C & \text{primitive constant} &\ \\
&\ \ \ \ \ \ |\ f s\ & \text{application} &\ \\
&\ \ \ \ \ \ |\ (s, t)\ & \text{pair} &\ \\
&\ \ \ \ \ \ |\ s\ \epsilon\ t\ & \text{additive conjunction} &\ \\
&\ \ \ \ \ \ |\ s\ \gamma\ t\ & \text{multiplicative disjunction} &\ \\
&\ \ \ \ \ \ |\ fst_⊗\ M\ & \text{first projection for multiplicative conjunction} &\ \\
&\ \ \ \ \ \ |\ snd_⊗\ M\ & \text{second projection for multiplicative conjunction} &\ \\
&\ \ \ \ \ \ |\ fst_{\&}\ M\ & \text{first projection for additive conjunction} &\ \\
&\ \ \ \ \ \ |\ snd_{\&}\ M\ & \text{second projection for additive conjunction} &\ \\
&\ \ \ \ \ \ |\ let\ (x, y) = d\ in\ e\ & \text{dependent pair pattern match} &\ \\
&\ \ \ \ \ \ |\ s \overset{π}{:} S & \text{type \& usage annotation} &\ \\
\end{align*}
\end{subfigure}
\end{figure}

Sorts $∗_i$ are explicitly leveled. Dependent function types, dependent conjunction types, and type annotations include a usage annotation $π$.

Judgements have the following form:

$$ x_1 \overset{ρ_1}{:} S_1, ..., x_n \overset{ρ_n}{:} S_n \vdash\ M \overset{σ}{:} T $$

where $ρ_1 ... ρ_n$ are elements of the semiring and $σ$ is either the $0$ or $1$ of the semiring. When $σ = 0$, it is the typing judgement. When $σ = 1$, the judgement expresses the computational resources required.

Further define the syntactic categories of usages $ρ, π$ and precontexts $Γ$:

\begin{align*}
ρ,π ∈ R \\
Γ := ⋄\ |\ Γ,x \overset{ρ}{:} S
\end{align*}

The symbol ⋄ denotes the empty precontext.

Precontexts contain usage annotations $ρ$ on constituent variables. Scaling a precontext, $πΓ$, is defined as follows:

\begin{align}
π(⋄)=⋄ \\
π(Γ,x \overset{ρ}{:} S) = πΓ,x \overset{πρ}{:} S
\end{align}

Usage annotations in types are not affected.

By the definition of a semiring, $0Γ$ sets all usage annotations to $0$.

Addition of two precontexts $Γ_1 + Γ_2$ is defined only when $0Γ_1 = 0Γ_2$:

\begin{align*}
⋄+⋄=⋄\\
(Γ_1,x \overset{ρ_1}{:} S) + (Γ_2,x \overset{ρ_2}{:} S) = (Γ_1+Γ_2), x \overset{ρ_1 + ρ_2}{:} S
\end{align*}

Contexts are identified within precontexts by the judgement $Γ\vdash$, defined by the following rules:

\begin{prooftree}
\AxiomC{}
\RightLabel{Emp}
\UnaryInfC{⋄\vdash}
\end{prooftree}

\begin{prooftree}
\AxiomC{$Γ \vdash$}
\AxiomC{$0Γ ⊢ S$}
\RightLabel{Ext}
\BinaryInfC{$Γ,x \overset{ρ}{:} S \vdash$}
\end{prooftree}

$0Γ ⊢ S$ indicates that $S$ is well-formed as a type in the context of $0Γ$.
$Emp$, for "empty", builds the empty context, and $Ext$, for "extend", extends a context $Γ$ with a new variable $x$ of type $S$ and usage annotation $ρ$.
All type formation rules yield judgements where all usage annotations in $Γ$ are $0$ — that is to say, type formation requires no computational resources).

Term judgements have the form:

\begin{align}
Γ ⊢ M \overset{σ}{:} S
\end{align}

where $σ \in {0,1}$. A judgement with $σ = 0$ constructs a term with no computational content, while a judgement with $σ = 1$ constructs a term which will be computed with. E.g.,

Typing judgement:
\begin{align*}
    n \overset{0}{:} Nat, x \overset{0}{:} Fin(n) ⊢ x \overset{0}{:} Fin(n)
\end{align*}
Computational judgement:
\begin{align*}
    n \overset{0}{:} Nat (?), x \overset{1}{:} Fin(n) ⊢ x \overset{1}{:} Fin(n)
\end{align*}

## Typing rules

### Variable & conversion rules

The variable rule selects an individual variable, type, and usage annotation from the context:

\begin{prooftree}
\AxiomC{$⊢ 0Γ,x \overset{σ}{:} S, 0Γ′$}
\RightLabel{Var}
\UnaryInfC{$0Γ,x \overset{σ}{:} S, 0Γ′ ⊢ x \overset{σ}{:} S$}
\end{prooftree}

The conversion rule allows conversion between judgementally equal types:

\begin{prooftree}
\AxiomC{$Γ ⊢ M \overset{σ}{:} S$}
\AxiomC{$0Γ ⊢ S≡T$}
\RightLabel{Conv}
\BinaryInfC{$Γ ⊢ M \overset{σ}{:} T$}
\end{prooftree}

Note that type equality is judged in a context with no resources.

### Primitive constants, functions & types

\begin{prooftree}
\AxiomC{$c \in C$}
\AxiomC{$\kappa \in K$}
\AxiomC{$c ⋮ \kappa$}
\RightLabel{Prim-Const}
\TrinaryInfC{$⊢ c \overset{σ}{:} \kappa$}
\end{prooftree}

Primitive constants are typed according to the primitive typing relation, and they can be produced in any computational quantity.

\begin{prooftree}
\AxiomC{$f \in F$}
\AxiomC{$f ⋮ (x \overset{π}{:} S) → T$}
\RightLabel{Prim-Fn}
\BinaryInfC{$⊢ f \overset{σ}{:} (x \overset{π}{:} S) → T$}
\end{prooftree}

Primitive functions are typed according to the primitive typing relation, and they can be produced in any computational quanitity.

Primitive functions can be dependently-typed.

\begin{prooftree}
\AxiomC{$f \in F$}
\AxiomC{$⊢ f \overset{σ}{:} (x \overset{π}{:} S) → T$}
\AxiomC{$Γ ⊢ M \overset{σ'}{:} S$}
\AxiomC{$σ′=0 ⇔ (π=0 ∨ σ=0)$}
\RightLabel{Prim-App}
\QuaternaryInfC{$πΓ ⊢ f M \overset{σ}{:} T[x := M]$}
\end{prooftree}

Applications of primitive functions impose the same usage constraints as regular abstractions.

### Dependent function types

Function types $(x \overset{π}{:} S) → T$ record usage of the argument. The formation rule is:

\begin{prooftree}
\AxiomC{$0Γ ⊢ S$}
\AxiomC{$0Γ,x \overset{0}{:} S ⊢ T$}
\RightLabel{Pi}
\BinaryInfC{$0Γ ⊢ (x \overset{π}{:} S) → T$}
\end{prooftree}

The usage annotation $π$ is not used in judgement of whether $T$ is a well-formed type. It is used
in the introduction and elimination rules to track how $x$ is used, and how to multiply the resources
required for the argument, respectively:

\begin{prooftree}
\AxiomC{$Γ,x \overset{σπ}{:} S ⊢ M \overset{σ}{:} T$}
\RightLabel{Lam}
\UnaryInfC{$Γ ⊢ λx.M \overset{σ}{:} (x \overset{π}{:} S) → T$}
\end{prooftree}

\begin{prooftree}
\AxiomC{$Γ_1 ⊢ M \overset{σ}{:} (x \overset{π}{:} S) → T$}
\AxiomC{$Γ_2 ⊢ N \overset{σ′}{:} S$}
\AxiomC{$0Γ_1 = 0Γ_2$}
\AxiomC{$σ′=0 ⇔ (π=0 ∨ σ=0)$}
\RightLabel{App}
\QuaternaryInfC{$Γ_1 + πΓ_2 ⊢ M N \overset{σ}{:} T[x := N]$}
\end{prooftree}

- $0Γ_1 = 0Γ_2$ means that $Γ_1$ and $Γ_2$ have the same variables with the same types
- In the introduction rule, the abstracted variable $x$ has usage $σπ$ so that non-computational production requires no computational input
- In the elimination rule, the resources required by the function and its argument, scaled to the amount required by the function, are summed
- The function argument $N$ may be judged in the 0-use fragment of the system if and only if we are already in the 0-use fragment ($σ = 0$) or the function will not use the argument ($π = 0$).

### Dependent multiplicative conjunction (tensor product)

Dependent tensor production formation rule:

\begin{prooftree}
\AxiomC{$0Γ ⊢ A$}
\AxiomC{$0Γ,x \overset{0}{:} S ⊢ T$}
\RightLabel{⊗}
\BinaryInfC{$0Γ ⊢ (x \overset{π}{:} S) ⊗ T$}
\end{prooftree}

\begin{prooftree}
\AxiomC{$0Γ ⊢$}
\RightLabel{I}
\UnaryInfC{$0Γ ⊢ I$}
\end{prooftree}

Type formation does not require any resources.

Introduction rule:

\begin{prooftree}
\AxiomC{$Γ_1 ⊢ M \overset{σ′}{:} S$}
\AxiomC{$0Γ_1 = 0Γ_2$}
\AxiomC{$Γ_2 ⊢ N \overset{σ}{:} T[x := M]$}
\AxiomC{$σ′=0 ⇔ (π=0 ∨ σ=0)$}
\QuaternaryInfC{$πΓ_1 + Γ_2 ⊢ (M,N) \overset{σ}{:} (x \overset{π}{:} S) ⊗ T$}
\end{prooftree}

\begin{prooftree}
\AxiomC{$0Γ ⊢$}
\UnaryInfC{$0Γ ⊢ ∗ \overset{σ}{:} I$}
\end{prooftree}

This is similar to the introduction rule for dependent function types above.

Elimination rule:

Under the erased ($σ=0$) part of the theory, projection operators can be used as normal:

\begin{prooftree}
\AxiomC{$Γ ⊢ M \overset{0}{:} (x \overset{π}{:} S) ⊗ T$}
\UnaryInfC{$Γ ⊢ fst_⊗\ M \overset{0}{:} A$}
\end{prooftree}

(should the last be $S$ ?)

\begin{prooftree}
\AxiomC{$Γ ⊢ M \overset{0}{:} (x \overset{π}{:} S) ⊗ T$}
\UnaryInfC{$Γ ⊢ snd_⊗\ M \overset{0}{:} T[x := fst_⊗(M)]$}
\end{prooftree}

Under the resourceful part:

\begin{prooftree}
\AxiomC{$0Γ_1, z \overset{0}{:} (x \overset{π}{:} S) ⊗ T ⊢ U$}
\AxiomC{$Γ_1 ⊢ M \overset{σ}{:} (x \overset{π}{:} S) ⊗ T$}
\AxiomC{$Γ_2, x \overset{σπ}{:} S, y \overset{σ}{:} T ⊢ N \overset{σ}{:} U[z := (x,y)]$}
\AxiomC{$0Γ_1 = 0Γ_2$}
\RightLabel{$⊗$ Elim}
\QuaternaryInfC{$Γ_1+Γ_2 ⊢ let\ (x,y) = M\ in\ N \overset{σ}{:} U[z := M]$}
\end{prooftree}

- Must pattern match out to ensure both parts of the product are used.

\begin{prooftree}
\AxiomC{$0Γ_1, x \overset{0}{:} I ⊢ U$}
\AxiomC{$Γ_1 ⊢ M \overset{σ}{:} I$}
\AxiomC{$Γ_2 ⊢ N \overset{σ}{:} U[x := *]$}
\AxiomC{$0Γ_1 = 0Γ_2$}
\RightLabel{$⊗$ Elim I}
\QuaternaryInfC{$Γ_1 + Γ_2 ⊢ let\ ∗ = M\ in\ N \overset{σ}{:} U[x := M]$}
\end{prooftree}

- Must explicitly eliminate elements of the unit type in the resourceful fragment.
- Simplifies to fst, snd in $σ=0$ fragment (should we combine the rules?)

- If we lambda-encoded pairs, is that isomorphic?

### Multiplicative disjunction

"Both separately in parallel"

Presumably cannot be dependent.

Should be able to provide guarantee of parallelism in low-level execution, both in bespoke & interaction net paths.

Formation rule:

\begin{prooftree}
\AxiomC{$Γ ⊢ (A \overset{σ}{:} S), (B \overset{σ'}{:} S')$}
\RightLabel{$\parr$}
\UnaryInfC{$Γ ⊢ (A \overset{σ}{:} S) \parr (B \overset{σ'}{:} S')$}
\end{prooftree}

Introduction rule:

\begin{prooftree}
\AxiomC{$Γ_1 ⊢ M \overset{σ}{:} S$}
\AxiomC{$0Γ_1 = 0Γ_2$}
\AxiomC{$Γ_2 ⊢ N \overset{σ}{:} T$}
\RightLabel{$\parr$-Intro}
\TrinaryInfC{$Γ_1 + Γ_2 ⊢ M\ \gamma\ N \overset{σ}{:} S \parr T$}
\end{prooftree}

do we need equality of 0-fragment?

Elimination rule:

\begin{prooftree}
\AxiomC{$Γ_1, M \overset{σ}{:} S ⊢ M' \overset{σ'}{:} S'$}
\AxiomC{$Γ_2, N \overset{σ}{:} T ⊢ N' \overset{σ'}{:} T'$}
\RightLabel{$\parr$-Elim}
\BinaryInfC{$Γ_1 + Γ_2 + M \gamma\ N \overset{σ}{:} S \parr T ⊢ (M', N') \overset{σ'}{:} (S ⊗ T)$}
\end{prooftree}

reformulate to be syntax-directed, we need some sort of "join"

### Additive conjunction

"Choose either"

Can be dependent.

Formation rule:

\begin{prooftree}
\AxiomC{$Γ ⊢ A \overset{σ}{∶} S$}
\AxiomC{$Γ ⊢ B \overset{σ}{:} T$}
\RightLabel{$\&$}
\BinaryInfC{$Γ ⊢ (A \overset{σ}{∶} S)\ \&\ (B \overset{σ'}{:} T)$}
\end{prooftree}

- Can we construct with $σ' /= σ$?

Introduction rule:

\begin{prooftree}
\AxiomC{$Γ_1 ⊢ M \overset{σ}{:} S$}
\AxiomC{$0Γ_1 = 0Γ_2$}
\AxiomC{$Γ_2 ⊢ N \overset{σ}{:} T[x := M]$}
\TrinaryInfC{$πΓ_1 + Γ_2 ⊢ M\ \epsilon\ N \overset{σ}{:} (x \overset{π}{:} S)\ \&\ T$}
\end{prooftree}

Elimination rules:

\begin{prooftree}
\AxiomC{$Γ ⊢ M\ \epsilon\ N \overset{σ}{:} (x \overset{π}{:} S)\ \&\ T$}
\UnaryInfC{$Γ ⊢ M \overset{πσ}{:} S$}
\end{prooftree}

\begin{prooftree}
\AxiomC{$Γ ⊢ M\ \epsilon\ N \overset{σ}{:} (x \overset{π}{:} S)\ \&\ T$}
\UnaryInfC{$Γ ⊢ N \overset{σ}{:} T[x := M]$}
\end{prooftree}

To-do: terms for elimination rules.

### Additive disjunction

"Might be either"

Presumably cannot be dependent.

Formation rules:

\begin{prooftree}
\AxiomC{$Γ ⊢ A \overset{σ}{∶} S$}
\RightLabel{$⊕_L$}
\UnaryInfC{$Γ ⊢ (A \overset{σ}{∶} S) ⊕ (B \overset{σ'}{:} S)$}
\end{prooftree}

\begin{prooftree}
\AxiomC{$Γ ⊢ B \overset{σ}{∶} S$}
\RightLabel{$⊕_R$}
\UnaryInfC{$Γ ⊢ (A \overset{σ'}{∶} S) ⊕ (B \overset{σ}{:} S)$}
\end{prooftree}

Do we get this for free with lambda-encoded datatypes?

### Universe (set type)

Let $S$ be a set of sorts ${i, j, k}$ with a total order.

Formation rule:

\begin{prooftree}
\AxiomC{$0Γ ⊢$}
\AxiomC{$i < j$}
\RightLabel{∗}
\BinaryInfC{$0Γ ⊢ ∗_i \overset{0}{:} ∗_j$}
\end{prooftree}

Introduction rule ($σ = 0$ fragment only):

\begin{prooftree}
\AxiomC{$0Γ ⊢ M \overset{0}{:} ∗_i$}
\AxiomC{$0Γ, x \overset{0}{:} M ⊢ N \overset{0}{:} ∗_i$}
\RightLabel{∗-Pi}
\BinaryInfC{$Γ ⊢ (x \overset{π}{:} M) → N \overset{0}{:} ∗_i$}
\end{prooftree}

### Self types

\begin{prooftree}
\AxiomC{$Γ, x: ιx.T ⊢ T : ∗_i$}
\RightLabel{Self}
\UnaryInfC{$Γ ⊢ ιx.T$}
\end{prooftree}

\begin{prooftree}
\AxiomC{$Γ ⊢ t : [x := t]T$}
\AxiomC{$Γ ⊢ ιx.T : ∗_i$}
\RightLabel{Self-Gen}
\BinaryInfC{$Γ ⊢ t : ιx.T$}
\end{prooftree}

\begin{prooftree}
\AxiomC{$Γ ⊢ t : ιx.T$}
\RightLabel{Self-Inst}
\UnaryInfC{$Γ ⊢ t : [x := t]T$}
\end{prooftree}

To-do for self types

- Syntax for self types?

### Equality judgements

Types are judgementally equal under beta reduction:

\begin{prooftree}
\AxiomC{$Γ ⊢ S$}
\AxiomC{$Γ ⊢ T$}
\AxiomC{$S →_{β} T$}
\RightLabel{≡-Type}
\TrinaryInfC{$Γ ⊢ S ≡ T$}
\end{prooftree}

Terms with the same type are judgementally equal under beta reduction:

\begin{prooftree}
\AxiomC{$Γ ⊢ M \overset{σ}{:} S$}
\AxiomC{$Γ ⊢ N \overset{σ}{:} S$}
\AxiomC{$M →_{β} N$}
\RightLabel{≡-Term}
\TrinaryInfC{$Γ ⊢ M ≡ N \overset{σ}{:} S$}
\end{prooftree}

To-do: do we have / need a rule for term equality?

### Sub-usaging

To-do: check if we can safely allow sub-usaging if the ring is the natural numbers, discuss here.

## Semantics

Contraction is $(λx.t : (π x : S) → T) s ⇝_{β} (t:T)[x := s:S]$.

The reflexive transitive closure of $⇝_{β}$ yields beta reduction $→_{β}$ as usual.

### Confluence

A binary relation $R$ has the diamond property iff. $∀ s p q . s R p ∧ s R q \implies ∃r . p R r ∧ q R r$.

### Parallel-step reduction

Parallel-step evaluation semantics a la McBride.

### Small-step reduction

Small-step evaluation semantics (so that one-step beta equality can be used in the theorem prover).

## Typechecking

Lay out syntax-directed typechecker following McBride's paper.

## Erasure

- Define erasure to untyped lambda calculus following McBride's paper.
- Define erasure to dependent elementary affine logic analogously (but not erasing the pi types).

Let programs of the untyped lambda calculus be $p ::= x\ |\ λx.p\ |\ p\ p\ |\ c$ where $c \in C$ is a primitive constant.

Define the erasure operator $▶$, such that erasure judgements take the form $Γ ⊢ t \overset{σ}{:} S \ ▶\  p$.

\begin{prooftree}
\AxiomC{$$}
\RightLabel{Prim-Erase}
\UnaryInfC{$c \ ▶\  c$}
\end{prooftree}

\begin{prooftree}
\AxiomC{$⊢ 0Γ,x \overset{σ}{:} S, 0Γ′$}
\RightLabel{Var-Erase-+}
\UnaryInfC{$0Γ,x \overset{σ}{:} S, 0Γ′ ⊢ x \overset{σ}{:} S \ ▶\  x$}
\end{prooftree}

\begin{prooftree}
\AxiomC{$t \overset{σ}{:} T \ ▶\  p$}
\AxiomC{$σπ = 0$}
\RightLabel{Lam-Erase-0}
\BinaryInfC{$λx.t : (x \overset{π}{:} S) → T \ ▶\  p$}
\end{prooftree}

\begin{prooftree}
\AxiomC{$t \overset{σ}{:} T \ ▶\  p$}
\AxiomC{$σπ /= 0$}
\RightLabel{Lam-Erase-+}
\BinaryInfC{$λx.t : (x \overset{π}{:} S) → T \ ▶\  λx.p$}
\end{prooftree}

\begin{prooftree}
\AxiomC{$Γ_1 ⊢ M \overset{σ}{:} (x \overset{π}{:} S) → T \ ▶\  p$}
\AxiomC{$Γ_2 ⊢ N \overset{0}{:} S$}
\AxiomC{$σπ = 0$}
\RightLabel{App-Erase-0}
\TrinaryInfC{$Γ_1 ⊢ M N \overset{σ}{:} T[x := N] \ ▶\  p$}
\end{prooftree}

\begin{prooftree}
\AxiomC{$Γ_1 ⊢ M \overset{σ}{:} (x \overset{π}{:} S) → T \ ▶\  p$}
\AxiomC{$Γ_2 ⊢ N \overset{σπ}{:} S \ ▶\  p'$}
\AxiomC{$σπ /= 0$}
\RightLabel{App-Erase-+}
\TrinaryInfC{$Γ_1 + Γ_2 ⊢ M N \overset{σ}{:} T[x := N] \ ▶\  p\ p'$}
\end{prooftree}

In the *Lam-Erase-0* rule, the variable $x$ bound in $t$ will not occur in the corresponding $p$, since it is bound with usage $0$, with which it will remain regardless of how the context splits, so the rule *Var-Erase-+* cannot consume it, only contemplate it.

Other terms erase as you would expect.

To-do:

- Constants $c \in C$ erase to themselves
- Functions $f \in F$ erase to themselves
- Function applications $f x$ of primitive $f \in F$ erase to $f (▶ x)$.

Computationally relevant terms are preserved, while terms which are only contemplated are erased.

## Examples

### Church-encoded natural numbers

- Representation
- Deriving induction for Church-encoded numerals using self-types

The Church-encoded natural $n$ can be typed as $⊢ λs.z.s {...} s z \overset{n}{:} (s \overset{π}{:} a → a) → (z \overset{1}{:} a) → a$ where $s$ is applied $n$ times.

### Linear lists

Example from McBride's paper:

$0List: (0X:∗)→∗$

$ωnil: (0X:∗)→ListX$

$ωcons: (0X:∗,1x:X,1xs:ListX)→ListX$

$ωlit: (0X:∗,0P: (0x:ListX)→∗)→(1n:P(nilX))→(ωc: (1x:X,0xs:ListX,1p:Pxs)→P(consX xxs))→(1xs:ListX)→P xs$

$ωappend:  (0X:∗,1xs:ListX,1ys:ListX)→ListX$

$append=λX.λxs.λys.litX(λ.ListX)ys(λx.λ.λxs′.consX xxs′)xs$

### Linear logic connectives

- Linear logic disjunctions & conjunctions
- Linear induction?