juvix/lab/docs

Proposal : MiniJuvix [draft]

% ---------- To typeset grammars -----------------------------
\renewcommand\.{\mathord.}
\newcommand{\EQ}{\mkern5mu\mathrel{::=}}
\newcommand{\OR}[1][]{\mkern17mu | \mkern12mu}
\newcommand{\Or}{\mathrel|}
\newcommand{\RT}[1]{\{#1\}}
\newcommand{\RV}[1]{\langle#1\rangle}
\newcommand{\Let}{\mathbf{let}\:}
\newcommand{\Q}{\mathrel|}
\newcommand{\I}{\color{blue}}
\newcommand{\O}{\color{green}}
% ---------- To typese rules ---------------------------------
\newcommand{\rule}[3]{%
\frac{%
\begin{gathered}\,{#2}%
\end{gathered}%
}{#3}\:{\mathsf{#1}}}
% Bidirectional judgements
\newcommand{\check}[4]{{{#1}\,\vdash\,{#2}\,\overset{{#3}}{\color{red}{\Leftarrow}}\, {#4}}}
\newcommand{\infer}[4]{{{#1}\,\vdash\,{#2}\,\overset{{#3}}{\color{blue}{\Rightarrow}}\, {#4}}}
% ------------------------------------------------------------

tags: Juvix

Abstract

MiniJuvix implements a programming language that takes variable resources very seriously in the programs. As mathematical foundation, we are inspired by Quantitative type theory (QTT), a dependent type theory that marriages ideas from linear logic and traditional dependently typed programs to writing memory-efficient programs using resource accounting. Among the language features, there is a type for a universe, dependent function types, tensor products, sum types, and more type formers.

The main purpose of MiniJuvix is to serve as a guide to supporting/extending the Juvix programming language, in particular, the design of a correct and efficient typechecker.

In this document we provide a work in progress report containing a description of the MiniJuvix bidirectional type checking algorithm. We have provide some Haskell sketches for the algorithm implementation.

The code will be available on the Github repository: heliaxdev/MiniJuvix. In this document, we only refer to the implementation provided in the qtt branch.

Language

Syntax

Quantities In the traditional QTT, each term has a usage/quantity annotation in the semiring \{0,1,\omega\}. Besides the semiring structure, we also consider different ordering of these terms. One choice for such an order states that 0,1 < \omega and 0 and 1 are not comparable, i.e., 0 \not < 1. This order is good, at least in the sense that terms of zero usage and 1 usage live in completely different worlds, from the evaluation point-of-view. Terms or zero usage are irrelevant at runtime, and we therefore erase them in the erasure phase. While, non-zero terms are present during compilation and execution of the program. We embrace this distinction in the implementation with the data type Relevant with constructors irrelevant and relevant.

\begin{aligned}
%x,y,z &\EQ \dotsb & \text{term variable} \\[.5em]
\pi,\rho,\sigma &\EQ 0 \Or 1 \Or \omega
  & \text{(quantity variables)} \\[.5em]
\end{aligned}

Judgements The core language in MiniJuvix is bidirectional syntax-directed, meaning that a judgement in the theory contains a term that is either checkable or inferable. A type judgement consists of a context, a term --the term quantity required--, a judgement mode, and a type. Precisely, the judgement mode is either checking or inferring, as illustrated in the rules below, respectively, using a red and blue arrow.

\begin{gathered}
\check{\Gamma}{t}{\sigma}{M}
\qquad
\infer{\Gamma}{t}{\sigma}{M}
\end{gathered}
%

We will refer to the erased part of the theory when the variable resource \sigma is zero in the type judgement ; otherwise, we are in the present segment of the theory. Another way to refer to this distinction is that no computation is required in the \sigma zero theory. Otherwise, we say that the judgement possess computation content.

Contexts The context in which a term is judged is fundamental to determine well-formed terms. Another name for context is environment. A context can be either empty or it can be extended by binding name annotations of the form $x \overset{\sigma}{:} M$ for a given type A.

\begin{aligned}
\Gamma &\EQ \emptyset \Or (\Gamma, x\overset{\pi}{:} A) & \text{(contexts)}
\end{aligned}

A needed context operation is scaling. The scalar product with a context \Gamma is defined by induction on the context structure. Given a scalar number \sigma, the product \sigma \cdot \Gamma denotes the context \Gamma after multiplying all its resources variables by \sigma.

\begin{aligned}
\color{green}{\sigma} \cdot \emptyset &:= \emptyset,\\
\color{green}{\sigma} \cdot (\Gamma, x \overset{\color{green}{\pi}}{:} A) &:= \color{green}{\sigma} \cdot \Gamma , x \overset{\color{green}{\sigma\cdot \pi}}{:} A.
\end{aligned}

The addition operation for contexts is a binary operation only defined between contexts with the same variable set. The latter condition is the proposition stating 0 \cdot \Gamma_1 = 0 \cdot \Gamma_2 between contexts \Gamma_1 and \Gamma_2. Consequently, adding contexts create another context with the same variables from the input but with their resource summed up.

\begin{aligned}
\emptyset+\emptyset &:=\emptyset \\
(\Gamma_{1}, x \overset{\color{green}{\sigma}}{:} A)+(\Gamma_{2}, x \overset{\color{green}{\pi}}{:} A) &:=(\Gamma_{1}+\Gamma_{2}), x^{\color{green}{\sigma+\pi}} S
\end{aligned}

Telescopes A resource telescope is the name for grouping types with resource information. We use telescopes in forming new types, for example, in forming new inductive types.

\begin{aligned} \Delta &\EQ () \Or \Delta(x \overset{\sigma}{:} A) & \text{(telescopes)} \end{aligned}

The \color{gray}{gray} cases below are expected to be incorporated in the future.

Types A type in the theory is one of the following synthactical cases.

\begin{aligned}
A , B%
&\EQ \mathcal{U} & \text{(universe type)} \\
&\OR (x \overset{\sigma}{:} A) \to B       &\text{(depend. fun. type)} \\
&\OR (x \overset{\sigma}{:} A) \otimes B   &\text{(tensor prod. type)} \\
&\OR A + B   &\text{(sum type)} \\
&\OR 1   &\text{(unit type)} \\
&\OR \color{gray}{P} &\color{gray}{\text{(primitive type)}}\\
&\OR \color{gray}{D} &\color{gray}{\text{(inductive type decl.)}}\\
&\OR \color{gray}{R} &\color{gray}{\text{(record type decl.)}}
\end{aligned}

On the other hand, we want to consider a set of primitive types, each of these with a set of primitive terms. An example of a primitive types is that of the type of boolens, denoted by \mathsf{Bool}. \mathsf{true} : \mathsf{Bool} and \mathsf{False} : \mathsf{Bool}.

Terms We refer to terms as those elements that can inhabit a type. So far, we have used as a term the metavariable x. A term can take one of the following shapes.

\begin{aligned}
u, v , t , f &\EQ \mathsf{Var}(x)    &\text{(variable)}\\
&\OR \mathsf{Ann}(x,A)           &\text{(type annotation)}\\
&\OR \mathsf{Lam}(x,t)           &\text{(lambda abstraction)}\\
&\OR\mathsf{App}(u,v)            &\text{(application)}\\
&\OR *                           &\text{(unit)}\\
&\OR \color{gray}{\mathsf{Fun}} &\color{gray}{\text{(named function)}}\\
&\OR \color{gray}{\mathsf{Con}} &\color{gray}{\text{(data constr.)}}
\end{aligned}

The explicit naming below like \mathsf{Ann} is on purpose. We want to avoid any confussion, for example, between type annotations and usage type annotation, i.e., x : A and x \overset{\sigma}{:} A.

Typing rules

We present the types rules almost in the same way as one would expect to see them in the implementation, i.e., using the bidirectional notation.

It must be assumed that contexts appearing in the rules are well-formed, i.e. terms build up using the following derivation rules.

\rule{\mathsf{empty}\mbox{-}\mathsf{ctx}}{
%
}
{\emptyset \ \mathsf{ctx}}
\qquad
\rule{,\mbox{-}\mathsf{ctx}}{
\Gamma \ \mathsf{ctx}
\qquad
\color{green}{\Gamma \vdash A \ \mathsf{type}}
\qquad
\sigma \ \mathsf{Quantity}
}
{
(\Gamma , x \overset{\sigma}{:} A) \ \mathsf{ctx}
}

\rule{\cdot\mbox{-}\mathsf{ctx}}{%
\Gamma \ \mathsf{ctx} \qquad \sigma \ \mathsf{Quantity}
}{
\sigma \cdot \Gamma \ \mathsf{ctx}
}\qquad
\rule{\mbox{+}\mbox{-}\mathsf{ctx}}{%
\Gamma_1 \ \mathsf{ctx} \qquad \Gamma_2 \ \mathsf{ctx} \qquad  0\cdot \Gamma_1 = 0 \cdot \Gamma_2
}{
\Gamma_1 + \Gamma_2 \ \mathsf{ctx}
}

Below, we describe the algorithms for checking and infering types in a mutually defined way. The corresponding algorithms are the functions check and infer in the implementation. The definition of each is the collection and interpretation of the typing rules (reading them bottom to top).

For example, the infer method is defined to work with three arguments: one implicit argument for the context \Gamma and two explicit arguments, respectively, the term t and its quantity \sigma. The output of the algorithm is precisely the type M for t in the rule below.

\begin{gathered}
\rule{}{
p_1 \cdots\ p_n
}{
\infer{\Gamma}{t}{\sigma}{M}
}
\end{gathered}
%

The variables p_i in the rule above are inner steps of the algorithm and the order in which they are presented matters. For example, an inner step can be infering a type, checking if a property holds for a term, reducing a term, or simply checking a term against another type.

A reduction step is denoted by $\Gamma \vdash t \rightsquigarrow t'$ or simply by t \rightsquigarrow t' whenever the context \Gamma is known. Such a reduction is obtained by calling eval in the implementation.

Remark. A term in the Core implementation is either a Checkable or Inferable term. We refer to these options as the mode of the term. In a typing rule the strategy/mode in a type judgement determines the mode of the term in the conclusion. In the example above, t is Inferable.

data Term : Set where
  Checkable : CheckableTerm → Term  -- terms with a type checkable.
  Inferable : InferableTerm → Term  -- terms that which types can be inferred.

TODO: we need to reflect on how we introduce the judgement \Gamma \vdash A\ \mathsf{type} of the well-formed types. This may change the way of presenting formation rules, as the first ones below.

Checking rules

This section mainly refers to the construction of checkable terms in the implementation.

Remark. We omit comments in the formation rules below. The general idea is that no resources are needed to form a type. Therefore, we only check when forming a type in the erase part of the theory, for both, premises and conclusion.

• UniverseType
• PiType
• Lam
• TensorType
• TensorIntro
• UnitType
• Unit
• SumType

Universe

Formation rule

\rule{}{\Gamma \ \mathsf{ctx} }{\Gamma \vdash \mathcal{U} \ \mathsf{type}}
\qquad
\begin{gathered}
\rule{Univ{\Leftarrow}}{
(0\cdot \Gamma) \ \mathsf{ctx}
}{
0\cdot \Gamma \vdash \mathcal{U}\, \overset{0}{\color{red}\Leftarrow}\,\, \mathcal{U}
}
\end{gathered}
%

Dependent function types

Formation rule

\rule{}{\Gamma \ \mathsf{ctx}
\qquad \Gamma \vdash A \ \mathsf{type}
\qquad (\Gamma , x \overset{\sigma}{:} A) \vdash B(x) \ \mathsf{type}
}{\Gamma \vdash (x \overset{\sigma}{:} A) \to B \ \ \mathsf{type}}
\\
\begin{gathered}
\rule{Pi{\Leftarrow}}{
0\cdot \Gamma \vdash A \, \overset{0}{\color{red}\Leftarrow}\,\mathcal{U}
\qquad
(0\cdot \Gamma,\,x\overset{0}{:}A)\vdash B \, \overset{0}{\color{red}\Leftarrow}\,\mathcal{U}\quad
}{
0\cdot \Gamma \vdash (x\overset{\pi}{:}A)\rightarrow B \overset{0}{\color{red}\Leftarrow}\mathcal{U}
}
\end{gathered}
%

Introduction rule

The lambda abstraction rule is the introduction rule of a dependent function type. The principal judgement in this case is the conclusion, and we therefore check against the corresponing type (x \overset{\sigma}{:} A) \to B. With the types A and B, all the information about the premise becomes known, and it just remains to check its judgement.

\begin{gathered}
\rule{Lam{\Leftarrow}}{
(\Gamma, x\overset{\sigma\pi}{:}A) \vdash b \,\overset{\sigma}{\color{red}\Leftarrow}\,B
}{
\Gamma \vdash \lambda x.b \overset{\sigma}{\color{red}\Leftarrow} (x\overset{\pi}{:}A)\rightarrow B
}
\end{gathered}
%

Another choice here is \lambda x.b instead of \lambda\,\mathsf{Ann}(x,A). b. The former option will change the strategy, to infer the conclusion, since one would have enough information for typing.

Tensor product types

Formation rule

\begin{gathered}
\rule{\otimes\mbox{-}{\Leftarrow}}{
0\cdot \Gamma \vdash A \,\overset{0}{\color{red}\Leftarrow}\,\mathcal{U}
\qquad
(0\cdot \Gamma, x\overset{0}{:}A) \vdash B\overset{0}{\color{red}\Leftarrow}\,\mathcal{U}\quad
}{
0\cdot \Gamma \vdash (x\overset{\pi}{:}A) \otimes B \overset{0}{\color{red}\Leftarrow}\,\mathcal{U}
}
\end{gathered}
%

Introduction rule

A rule to introduce pairs in QTT appears in Section 2.1.3 in Atkey's paper. We here present this rule in a more didactical way but also following the bidirectional recipe. Briefly, the known rule is splitted in two cases, the erased and present part of the theory, after studying the usage variable in the conclusion. Recall that forming pairs is the way one introduces values of the tensor product. One then must check the rule conclusion. After doing this, the types A and B become known facts and it makes sense to check the types in the premises. The usage bussiness follows a similar reasoning as infering applications.

\begin{gathered}
\rule{\otimes I{\Leftarrow}}{
\check{\sigma\pi \cdot \Gamma_1}{u}{\sigma\pi}{A}
\qquad
\check{\Gamma_2}{v}{\sigma}{B[u/x]}\quad
\color{gray}{0 \cdot \Gamma_1 = 0\cdot \Gamma_2}\quad
}{
\check{\sigma\pi\cdot \Gamma_1 + \Gamma_2}{(u,v)}{\sigma}{(x\overset{\pi}{:}A) \otimes B}
}
\end{gathered}
%

Essentially, we are forming \sigma dependent pairs where the first cordinate, u, is used $\pi$ times in the second component. This is the reason for having $\sigma\pi\cdot \Gamma_1$ in the conclusion since, u is taken from \Gamma_1. The gray premises below are necessary, since one must ensure that the addition between context is possible.

Finally, we obtain the following two rules that make up the original one.

1. \begin{gathered}
   

\rule{\otimes I_1{\Leftarrow}}{ \color{green}{\sigma\pi = 0} \qquad 0\cdot \Gamma \vdash u ,\overset{0}{\color{red}\Leftarrow},A \qquad \Gamma \vdash v ,\overset{\sigma}{\color{red}\Leftarrow},B[u/x] \qquad }{ \Gamma \vdash (u,v)\overset{\sigma}{\color{red}\Leftarrow} (x\overset{\pi}{:}A) \otimes B } \end{gathered} %

2. $$\begin{gathered}
\rule{\otimes I_2{\Leftarrow}}{
\color{green}{\sigma\pi \neq 0}\qquad
\Gamma_{1} \vdash u \,\overset{1}{\color{red}\Leftarrow}\,A
\qquad
\Gamma_{2} \vdash v \,\overset{\sigma}{\color{red}\Leftarrow}\,B[u/x]
\qquad
\color{gray}{0 \cdot \Gamma_1 = 0\cdot \Gamma_2}\quad
}{
\color{green}{\sigma\pi}\cdot \Gamma_{1}+\Gamma_{2} \vdash (u,v)\overset{\sigma}{\color{red}\Leftarrow} (x\overset{\pi}{:}A) \otimes B
}
\end{gathered}
%

Unit type

\rule{}{
0 \cdot \Gamma \ \mathsf{ctx}
}{
\Gamma \vdash 1 \ \mathsf{type}
}
\qquad
\rule{1\mbox{-}I}{
0 \cdot \Gamma \ \mathsf{ctx}
}{
\check{0\cdot\Gamma}{1}{0}{\mathcal{U}}
}
\qquad
\rule{*\mbox{-}I}{
0 \cdot \Gamma \ \mathsf{ctx}
}{
\check{0\cdot\Gamma}{*}{0}{1}
}

Sum type

TODO

Inductive types

TODO

Conversion rules

Include the rules for definitional equality:

• β-equality,
• reflexivity,
• symmetry,
• transitivity, and
• congruence.

\begin{gathered}
\rule{conv{\Leftarrow}}{
\Gamma \vdash M \,\overset{\sigma}{\color{blue}\Rightarrow}\,S \qquad
\Gamma \vdash S\, \overset{0}{\color{red}\Leftarrow}\, \mathcal{U}\qquad
\Gamma \vdash T \,\overset{0}{\color{red}\Leftarrow}\,\mathcal{U} \qquad
\color{green}{S =_{\beta} T}\ \,\,\,
}{
\Gamma \vdash M \overset{\sigma}{\color{red}\Leftarrow} T
}
\end{gathered}
%

Type inference

The algorithm that implements type inference is called infer. Inspired by Agda and its inference strategy, MiniJuvix only infer values that are uniquely determined by the context. There are no guesses. Either we fail or get a unique answer, giving us a predicatable behaviour.

By design, a term is inferable if it is one of the following cases.

• Variable
• Annotation
• Application
• Tensor type elim
• Sum type elim

Each case above has as a rule in what follows.

The Haskell type of infer would be similar as the following.

infer :: Quantity -> InferableTerm -> Output (Type , Resources)

where

Output = Either ErrorType
Resources = Map Name Quantity

Variable

A variable can be free or bound. If the variable is free, the rule is as follows.

Free variable

\begin{gathered}
\rule{Var⇒}{
(x :^{\sigma} M) \in \Gamma
}{
  \Gamma \vdash \mathsf{Free}(x) {\color{blue}\Rightarrow}^{\sigma} M
}
\end{gathered}
%

Explanation:

1. The input to infer is a variable term of the form Free x.
2. The only case for introducing a variable is to have it in the context.
3. Therefore, we ask if the variable is in the context.
4. If it's not the case, throw an error.
5. Otherwise, one gets a hypothesis x :^\sigma S from the context that matches x.
6. At the end, we return two things: 6.1. first, the inferred type and 6.2. a table with the new usage information for each variable.

Haskell prototype:

infer σ (Free x) = do
  Γ <- asks contextMonad
  case find ((== x) . getVarName) Γ of
    Just (BindingName _ _σ typeM)
      -> return (typeM, updateResources (x, _σ) )
    Nothing
      -> throwError "Variable not present in the context"

The method updateResources rewrites the map tracking names with their quantities.

Bound variables

The case of theBound variable throws an error.

Annotations

\begin{gathered}
\rule{Ann{⇒}}{
0\cdot \Gamma \vdash A\,{\color{red}\Leftarrow}^0\,\mathcal{U}
\qquad
\Gamma \vdash x\,{\color{red}\Leftarrow}^\sigma\, A
}{
  \Gamma \vdash \mathsf{Ann}(x,A)\,{\color{blue}\Rightarrow}^{\sigma}\,A
}
\end{gathered}
%

Any annotation possess type information that counts as known facts, and we therefore infer. However, this is a choice.

• First, we must check that A is a type, i.e., a term in some universe. Because there is only one universe we denote it by \mathcal{U}. The formation rule for types has no computation content, then the usage is zero in this case.

• Second, the term x needs to be checked against A using the same usage \sigma we need in the conclusion. The context for this is \Gamma. There is one issue here. This type checking expects A to be in normal form. When it is not, typechecking the judgement \Gamma \vdash x \Leftarrow^\sigma A may give us a false negative.

  • Example: Why do we need A'? Imagine that we want to infer the type of v given \Gamma \vdash x : \mathsf{Ann}(v, \mathsf{Vec}(\mathsf{Nat},2+2)). Clearly, the answer should be Vec(Nat,4). However, this reasoning step requires computation. $$\Gamma \vdash x : \mathsf{Ann}(v, \mathsf{Vec}(\mathsf{Nat},2+2)) \Rightarrow \mathsf{Vec}(\mathsf{Nat},4)),.$$

• Using M' as the normal form of A, it remains to check if x is of type A'. If so, the returning type is A' and the table resources has to be updated (the \color{gray}{gray} \Theta in the rule below).

\begin{gathered}
\rule{Ann{⇒}}{
\check{0\cdot \Gamma}{A}{0}{\mathcal{U}}
\qquad
A \color{green}{\rightsquigarrow} A'
\qquad
\check{\Gamma}{x}{\sigma}{A'} \color{darkgrey}{\dashv \Theta}
}{
\infer{\Gamma}{\mathsf{Ann}(x,A)}{\sigma}{A'}
\color{darkgrey}{\dashv \Theta}
}
\end{gathered}
%

Haskell prototype:

infer _ (Ann termX typeM) = do
  _         <- check (0 .*. context) typeM zero Universe
  typeM'    <- evalWithContext typeM
  (_ , newUsages) <- check context termX typeM'
  return (typeM' , newUsages)

Application

Elimination rule

Recall the task is to find A in $\Gamma \vdash \mathsf{App}(f,x) :^{\sigma} A$. If we follow the bidirectional type-checking recipe, then it makes sense to infer the type for an application, i.e., $\Gamma \vdash \mathsf{App}(f,x) \Rightarrow^{\sigma} A$. An application essentially removes a lambda abstraction introduced earlier in the derivation tree. The rule for this inference case is a bit more settle, especially because of the usage variables.

To introduce the term of an application, \mathsf{App}(f,x), it requires to give/have a judgement saying that f is a (dependent) function, i.e., $\Gamma \vdash f \overset{\sigma}{:} (x \overset{\pi}{:} A) \to B$, for usages variables \sigma and \pi. Then, given \Gamma, the function f uses \pi times its input, mandatory. We therefore need \sigma\pi resources of an input for f if we want to apply f \sigma times, as in the conclusion $\Gamma \vdash \mathsf{App}(f,x) \Rightarrow^{\sigma} A$.

In summary, the elimination rule is often presented as follows.

\begin{gathered}
\rule{}{
\Gamma \vdash f :^{\sigma} (x : ^\pi A) \to  B
\qquad
\sigma\pi\cdot\Gamma' \vdash x : ^{\sigma\pi} A
}{
\Gamma + \sigma\pi\cdot\Gamma'  \vdash \mathsf{App}(f,x)  :^{\sigma} B
}
\end{gathered}
%

The first judgement about f is principal. Then, it must be an inference step. After having inferred the type of f, the types A and B become known facts. It is then correct to check the type of x against A.

\begin{gathered}
\rule{}{
\Gamma \vdash f {\color{blue}\Rightarrow}^{\sigma}(x : ^\pi A) \to  B
\qquad
\sigma\pi\cdot\Gamma' \vdash x {\color{red}\Leftarrow}^{\sigma\pi} A
\qquad
\color{gray}{0 \cdot \Gamma = 0 \cdot \Gamma'}
}{
\Gamma + \sigma\pi\cdot\Gamma'  \vdash \mathsf{App}(f,x) \,{\color{blue}\Rightarrow^{\sigma}}\, B
}
\end{gathered}
%

To make our life much easier, the rule above can be splitted in two cases, emphasising the usage bussiness.

1. \begin{gathered}
   

\rule{App{\Rightarrow_1}}{ \color{green}{\sigma \cdot \pi = 0} \qquad \Gamma \vdash f {\color{blue}\Rightarrow^{\sigma}} (x :^{\pi} A) \to B \qquad 0\cdot \Gamma \vdash x {\color{red}\Leftarrow^{0}} A \qquad }{ \infer{\Gamma}{\mathsf{App}(f,x)}{\sigma}{B} } \end{gathered} %

2. $$\begin{gathered}
\rule{App{\Rightarrow_2}}{
\color{green}{\sigma \cdot \pi \neq 0}
\qquad
\infer{\Gamma_1}{f}{\sigma}{(x :^{\pi} A) \to B}
\qquad
\check{\Gamma_2}{x}{1}{A}
\qquad
\color{gray}{0 \cdot \Gamma_1 = 0 \cdot \Gamma_2}
\quad
}{
\infer{\Gamma_1 + \sigma \pi\cdot \Gamma_2}{\mathsf{App}(f,x)}{\sigma}{B}
}
\end{gathered}

In summary, we infer the type of f. If it is a $\Pi$-type, then one checks whether \sigma\pi is zero or not. If so, we use Rule No.1, otherwise, Rule No. 2. Otherwise, something goes wrong, an error arise.

Sketch:

infer σ (App f x) = do
  (arrowAtoB, usages) <- infer σ f
  case arrowAtoB of
    IsPiType π _ typeA typeB -> do
      σπ <- case (σ .*. π) of
       -- Rule No. 1
       Zero -> do
         (_ , nqs) <- check x typeA (mult Zero context)
          return nqs
       -- Rule No. 2
       _ -> undefined -- TODO (mult σπ context)
    -- f is not a function:
    ty -> throwError $ Error ExpectedPiType ty (App f x)

In the rules above, we have the lemma:

• 1 \cdot \Gamma \vdash x :^1 A entails that $\sigma \cdot \Gamma \vdash x :^\sigma A$ for any usage \sigma.

Tensor products

Elimination rule

In Atkey's QTT, there is no $\Sigma$-types but instead tensor product types. As with any other elimination rule, in the principal judgement, we synthetise a type. In our case, the principal judgement shows up in the first premise, which is the fact that M is a tensor product type. If we infer that, the types $A$ and B become known facts. Then, the type C, depending on A and B become checkable, also making the next judgement checking.

\begin{gathered}
\rule{TensorElim{\Rightarrow}}{
\infer{\Gamma_{1}}{M}{\sigma}{(x\overset{\pi}{:}A)\otimes B}
\\
\check{(0\cdot \Gamma_{1},z\overset{0}{:}(x\overset{\pi}{:}A)\otimes B)}{C}{0}{\mathcal{U}}
\\
\check{(\Gamma_{2}, u \overset{\sigma\pi}{:} A, v\overset{\sigma}{:}B)}{%
N}{\sigma}{C[(x,y)/z]}
}{
\Gamma_{1}+\Gamma_{2} \vdash \mathsf{let}\,z@(u,v)=M\,\,\mathsf{in}\,\,N :C \overset{\sigma}{\color{blue}\Rightarrow}\, C[M/x]
}
\end{gathered}
%

Remark Inspired by the tensor product rules in linear logic, there is a need to decompose a pair in its components. We have to be sure that all the resources in each component are effectively used. This mechanism needs to be introduced somewhere somehow, Idk yet. It is the keyword \mathsf{let}\mbox{-}\mathsf{in}.

Sum type elim

TODO

References

• Robert Atkey. 2018. Syntax and Semantics of Quantitative Type Theory. In Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS '18). Association for Computing Machinery, New York, NY, USA, 56–65. DOI:https://doi.org/10.1145/3209108.3209189
• Jana Dunfield and Neel Krishnaswami. 2021. Bidirectional Typing. ACM Comput. Surv. 54, 5, Article 98 (June 2022), 38 pages. DOI:https://doi.org/10.1145/3450952
• James Wood, Robert Atkey. A Linear Algebra Approach to Linear Metatheory. Arxiv: https://arxiv.org/abs/2005.02247
• Andy Morris. Juvix: Core language documentation. https://juvix.readthedocs.io/en/latest/compiler/core/core-language.html#id6
• Andy Morris. Proposal: Records in Core. https://hackmd.io/UT269VN1R6-qchHaWzg7KQ
• SVOBODA, Tomáš. Additive Pairs in Quantitative Type Theory. Praha, 2021. Diplomová práce Univerzita Karlova, Matematicko-fyzikální fakulta, Katedra algebry. Vedoucí práce Šefl, Vít. http://hdl.handle.net/20.500.11956/127263
• Conor McBride. 2016. I Got Plenty o' Nuttin'. In A List of Successes That Can Change the World-Essays Dedicated to Philip Wadler on the Occasion of His 60 t h Birthday. 207-233. DOI:https://doi.org/10.1007/978-3-319-30936-1