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added new rules
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docs/README.md
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docs/README.md
@ -39,18 +39,19 @@ $$
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MiniJuvix implements a programming language that takes variable resources very
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seriously in the programs. As mathematical foundation, we are inspired by
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Quantitative type theory (QTT), a dependent type theory that marriages ideas
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from linear logic and traditional dependently typed programs to writing memory-
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efficient programs using resource accounting. Some of the fincludes a universe,
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dependent function types, tensor products, sum types, among others. The main
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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,
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dependent function types, tensor products, sum types, among others.
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The main
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purpose of MiniJuvix is to serve as a guide to supporting/extending the
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[Juvix](/Q5LbuHI5RXaJ8mD08yW7-g) programming language, in particular, the design
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of its typechecker and the addition of new features.
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We provide a work in progress report containing a description of the MiniJuvix
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bidirectional type checking algorithm. Haskell sketches are provided on the
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implementation of the algorithm.
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In this document we provide a work in progress report containing a description
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of the MiniJuvix bidirectional type checking algorithm. Haskell sketches are
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provided on the implementation of the algorithm.
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**Sources** The code is be available on the Github repository:
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The code is be available on the Github repository:
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[heliaxdev/MiniJuvix](https://github.com/heliaxdev/MiniJuvix). In this document,
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we refer to the implementation provided in the `qtt` branch.
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@ -110,13 +111,14 @@ computation is required in the $\sigma$ zero case. Otherwise, the judgement
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possess *computation content*.
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**Contexts** The contexts we judge a term are fundamental. A context can be either empty or
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it can be extended by binding name annotations of the form $x
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**Contexts** The context in which a term is judged is fundamental to determine
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well-formed terms. Another name for context is *environment*. A context can be
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either empty or it can be extended by binding name annotations of the form $x
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\overset{\sigma}{:} M$ for a given type $A$.
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$$
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\begin{aligned}
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\Gamma &\EQ \emptyset \Or \Gamma, x :^{\sigma} A & \text{(contexts)}
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\Gamma &\EQ \emptyset \Or (\Gamma, x\overset{\pi}{:} A) & \text{(contexts)}
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\end{aligned}
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$$
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@ -125,9 +127,23 @@ context $\Gamma$ is defined by induction on the structure of the contexts. Given
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a scalar number $\sigma$, the product $\sigma \cdot \Gamma$ denotes the context
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after multiplying *all* resources variables by $\sigma$.
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The addition of context is a binary operation between contexts with the same
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variable set. A context addition creates a new context which has all the
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variables from the given context but with the resource varaibles summed up.
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$$
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\begin{aligned}
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\sigma \cdot \emptyset &:= \emptyset,\\
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\sigma \cdot (\Gamma, x \overset{\pi}{:} A) &:= \sigma \cdot \Gamma , x \overset{\sigma\pi}{:} A.
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\end{aligned}
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$$
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The addition of context is a binary operation only defined between contexts with
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the same variable set. This condition is the proposition $0 \cdot \Gamma_1 = 0 \cdot \Gamma_2$ between the contexts $\Gamma_1$ and $\Gamma_2$. Then, adding contexts create another one with all the
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initial variables from the input with the resource varaibles summed up, as follows.
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$$
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\begin{aligned}
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\emptyset+\emptyset &:=\emptyset \\
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(\Gamma_{1}, x \overset{\sigma}{:} A)+(\Gamma_{2}, x \overset{\pi}{:} A) &:=(\Gamma_{1}+\Gamma_{2}), x^{\sigma+\pi} S
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\end{aligned}
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$$
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**Telescopes** A *resource* telescope is the name for grouping types with some resource
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variable information. We use telescopes in forming new types below, for example,
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@ -185,20 +201,16 @@ i.e., $x : A$ and $x \overset{\sigma}{:} A$.
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# Typing rules
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We present the types rules in the same way as one would expect to see in the
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implementation using the bidirectional approach. Then, even though when type
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checking and type inference are mutually definied algorithms, checking goes
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first and then inference, similarly, as with the following definition. See the corresponding file
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[Core.agda](https://github.com/heliaxdev/MiniJuvix/blob/qtt/src/MiniJuvix/Syntax/Core.agda).
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implementation using the bidirectional approach. We must also assume that the contexts appearing in the rules are *well-formed*.
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```haskell
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data Term : Set where
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Checkable : CheckableTerm → Term -- terms with a type checkable.
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Inferable : InferableTerm → Term -- terms that which types can be inferred.
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{-# COMPILE AGDA2HS Term #-}
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```
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TODO: defined what means well-formed for contexts.
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Checking and type inference are the functions `check` and `infer` in the
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implementation. For example, the `infer` method is defined to work with three
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Below, we describe the algorithms for checking and infering types in a mutually
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defined way. The corresponding algorithms are the functions `check` and `infer`
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in the implementation. The definition of each is the collection and
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interpretation of the typing rules (read them bottom-up) given in this section.
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For example, the `infer` method is defined to work with three
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arguments: one implicit argument for the context $\Gamma$ and two explicit
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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.
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@ -207,7 +219,7 @@ $$
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\rule{}{
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p_1 \cdots\ p_n
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}{
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\Gamma \vdash t \Rightarrow^\sigma M
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\infer{\Gamma}{t}{\sigma}{M}
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}
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\end{gathered}
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%
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@ -220,6 +232,38 @@ type. A *reduction* step is denoted by $\Gamma \vdash t \rightsquigarrow t'$ or
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simply by $t \rightsquigarrow t'$ whenever the context $\Gamma$ is known. Such a
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reduction is obtained by calling `eval` in the implementation.
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*Remark** A term in the
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[Core](https://github.com/heliaxdev/MiniJuvix/blob/qtt/src/MiniJuvix/Syntax/Core.agda) implementation is either a checkable or inferable term. In a typing rule
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the is no explicit reference to the term kind. However, that information becomes
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evident after the given judgement mode in which the term is introduced/judged.
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```haskell
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data Term : Set where
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Checkable : CheckableTerm → Term -- terms with a type checkable.
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Inferable : InferableTerm → Term -- terms that which types can be inferred.
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```
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### Well-formed context
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$$
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\rule{\mathsf{empty}}{
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%
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}
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{\emptyset \ \mathsf{ctx}}
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\qquad
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\rule{\mathsf{ext}}{
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\Gamma \ \mathsf{ctx}
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\qquad
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\Gamma \vdash A \ \mathsf{type}
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}
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{
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(\Gamma , x \overset{\sigma}{:} A) \ \mathsf{ctx}
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}
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$$
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**TODO**: need to reflect on how to introduce the judgement $\Gamma \vdash A\ \mathsf{type}$, the well-formed types. This may change the way of presenting formation rules, as the first ones below.
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## Checking rules
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This section mainly refers to the construction of checkable terms in the
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@ -236,8 +280,8 @@ conclusion.
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- [x] Lam
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- [x] TensorType
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- [x] TensorIntro
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- [ ] UnitType
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- [ ] Unit
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- [x] UnitType
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- [x] Unit
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- [ ] SumType
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@ -246,9 +290,12 @@ conclusion.
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**Formation rule**
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$$\begin{gathered}
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$$
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\rule{}{\Gamma \ \mathsf{ctx} }{\Gamma \vdash \mathcal{U} \ \mathsf{type}}
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\qquad
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\begin{gathered}
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\rule{Univ{\Leftarrow}}{
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0\cdot \Gamma \vdash
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0\cdot \Gamma \ \mathsf{ctx}
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}{
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0\cdot \Gamma \vdash \mathcal{U}\, \overset{0}{\color{red}\Leftarrow}\,\, \mathcal{U}
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}
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@ -260,11 +307,17 @@ $$
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**Formation rule**
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$$\begin{gathered}
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$$
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\rule{}{\Gamma \ \mathsf{ctx}
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\qquad \Gamma \vdash A \ \mathsf{type}
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\qquad (\Gamma , x \overset{\sigma}{:} A) \vdash B(x) \ \mathsf{type}
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}{\Gamma \vdash (x \overset{\sigma}{:} A) \to B \ \ \mathsf{type}}
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\\
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\begin{gathered}
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\rule{Pi{\Leftarrow}}{
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0\cdot \Gamma \vdash A \, \overset{0}{\color{red}\Leftarrow}\,\mathcal{U}
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\qquad
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0\cdot \Gamma,\,x\overset{0}{:}A \vdash B \, \overset{0}{\color{red}\Leftarrow}\,\mathcal{U}\quad
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(0\cdot \Gamma,\,x\overset{0}{:}A)\vdash B \, \overset{0}{\color{red}\Leftarrow}\,\mathcal{U}\quad
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}{
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0\cdot \Gamma \vdash (x\overset{\pi}{:}A)\rightarrow B \overset{0}{\color{red}\Leftarrow}\mathcal{U}
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}
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@ -284,7 +337,7 @@ becomes known, and it just remains to check its judgment.
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$$\begin{gathered}
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\rule{Lam{\Leftarrow}}{
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\Gamma, x\overset{\sigma\pi}{:}A \vdash M \,\overset{\sigma}{\color{red}\Leftarrow}\,B
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(\Gamma, x\overset{\sigma\pi}{:}A) \vdash M \,\overset{\sigma}{\color{red}\Leftarrow}\,B
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}{
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\Gamma \vdash \lambda x.M \overset{\sigma}{\color{red}\Leftarrow} (x\overset{\pi}{:}A)\rightarrow B
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}
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@ -300,7 +353,7 @@ $$\begin{gathered}
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\rule{\otimes\mbox{-}{\Leftarrow}}{
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0\cdot \Gamma \vdash A \,\overset{0}{\color{red}\Leftarrow}\,\mathcal{U}
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\qquad
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0\cdot \Gamma, x\overset{0}{:}A \vdash B\overset{0}{\color{red}\Leftarrow}\,\mathcal{U}
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(0\cdot \Gamma, x\overset{0}{:}A) \vdash B\overset{0}{\color{red}\Leftarrow}\,\mathcal{U}\quad
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}{
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0\cdot \Gamma \vdash (x\overset{\pi}{:}A) \otimes B \overset{0}{\color{red}\Leftarrow}\,\mathcal{U}
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}
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@ -314,11 +367,10 @@ A rule to introduce pairs in QTT appears in [Section 2.1.3 in Atkey's
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paper](https://bentnib.org/quantitative-type-theory.pdf). We do present the same
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rule in a more a more didactical way but also following the bidirectional
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recipe. Briefly, the known rule is splitted in two cases, the erased and present
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part of the theory, i.e., respectively, looking at the conclusion. Recall that
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forming pairs is the way one introduces values of the tensor product. Then, one
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must check the conclusion. After doing this, the types $A$ and $B$ become known
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facts and it makes sense to check the types in the premises. The usage bussiness
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follows the similar reasoning given for applications.
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part of the theory, after studying the usage variable in the conclusion. Recall
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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
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known facts and it makes sense to check the types in the premises. The usage
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bussiness follows a similar reasoning as infering applications.
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$$\begin{gathered}
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@ -374,6 +426,35 @@ $$
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%
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$$
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### Unit type
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$$
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\rule{}{
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0 \cdot \Gamma \ \mathsf{ctx}
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}{
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\Gamma \vdash 1 \ \mathsf{type}
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}
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\qquad
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\rule{1\mbox{-}I}{
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0 \cdot \Gamma \ \mathsf{ctx}
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}{
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\check{0\cdot\Gamma}{1}{0}{\mathcal{U}}
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}
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\qquad
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\rule{*\mbox{-}I}{
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0 \cdot \Gamma \ \mathsf{ctx}
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}{
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\check{0\cdot\Gamma}{*}{0}{1}
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}
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$$
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### Sum type
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TODO
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### Inductive types
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TODO
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## Conversion rules
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@ -482,11 +563,11 @@ The case of the`Bound` variable throws an error.
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$$
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\begin{gathered}
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\rule{Ann{⇒}}{
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0\cdot \Gamma \vdash M\,{\color{red}\Leftarrow}^0\,\mathcal{U}
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0\cdot \Gamma \vdash A\,{\color{red}\Leftarrow}^0\,\mathcal{U}
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\qquad
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\Gamma \vdash x\,{\color{red}\Leftarrow}^\sigma\, M
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\Gamma \vdash x\,{\color{red}\Leftarrow}^\sigma\, A
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}{
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\Gamma \vdash \mathsf{Ann}(x,M)\,{\color{blue}\Rightarrow}^{\sigma}\,M
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\Gamma \vdash \mathsf{Ann}(x,A)\,{\color{blue}\Rightarrow}^{\sigma}\,A
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}
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\end{gathered}
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%
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@ -494,33 +575,33 @@ $$
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Any annotation possess type information that counts as known facts, and we therefore infer. However, this is a choice.
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- First, we must check that $M$ is a type, i.e., a term in *some* universe.
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- First, we must check that $A$ is a type, i.e., a term in *some* universe.
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Because there is only one universe we denote it by $\mathcal{U}$. The formation
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rule for types has no computation content, then the usage is zero in this case.
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- Second, the term $x$ needs to be checked against $M$ using the same usage
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- Second, the term $x$ needs to be checked against $A$ using the same usage
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$\sigma$ we need in the conclusion. The context for this is $\Gamma$. There is
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one issue here. This type checking expects $M$ to be in normal form. When it is
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not, typechecking the judgment $\Gamma \vdash x \Leftarrow^\sigma M$ may give us
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one issue here. This type checking expects $A$ to be in normal form. When it is
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not, typechecking the judgment $\Gamma \vdash x \Leftarrow^\sigma A$ may give us
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a false negative.
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- *Example*: Why do we need $M'$? 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)`.
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- *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)`.
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However, this reasoning step requires computation. $$\Gamma \vdash x : \mathsf{Ann}(v, \mathsf{Vec}(\mathsf{Nat},2+2)) \Rightarrow \mathsf{Vec}(\mathsf{Nat},4))\,.$$
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- Using $M'$ as the normal form of $M$, it remains to check if $x$ is of type
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$M'$. If so, the returning type is $M'$ and the resources map has to be updated
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- Using $M'$ as the normal form of $A$, it remains to check if $x$ is of type
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$A'$. If so, the returning type is $A'$ and the table resources has to be updated
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(the $\color{gray}{gray}$ $\Theta$ in the rule below).
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$$
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\begin{gathered}
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\rule{Ann{⇒}}{
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0\cdot \Gamma \vdash M \Leftarrow^0 \mathcal{U}
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\check{0\cdot \Gamma}{A}{0}{\mathcal{U}}
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\qquad
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M \rightsquigarrow M'
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A \color{green}{\rightsquigarrow} A'
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\qquad
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\Gamma \vdash x \Leftarrow^\sigma M' \color{darkgrey}{\dashv \Theta}
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\check{\Gamma}{x}{\sigma}{A'} \color{darkgrey}{\dashv \Theta}
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}{
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\Gamma \vdash \mathsf{Ann}(x,M) \Rightarrow^{\sigma} M'
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\color{darkgrey}{\dashv \Theta}
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\infer{\Gamma}{\mathsf{Ann}(x,A)}{\sigma}{A'}
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\color{darkgrey}{\dashv \Theta}
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}
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\end{gathered}
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%
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@ -540,20 +621,20 @@ infer _ (Ann termX typeM) = do
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**Elimination rule**
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Recall the task is to find $M$ in $\Gamma \vdash \mathsf{App}(f,x) :^{\sigma}
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M$. If we follow the bidirectional type-checking recipe, then it makes sense to
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Recall the task is to find $A$ in $\Gamma \vdash \mathsf{App}(f,x) :^{\sigma}
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A$. If we follow the bidirectional type-checking recipe, then it makes sense to
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infer the type for an application, i.e., $\Gamma \vdash \mathsf{App}(f,x)
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\Rightarrow^{\sigma} M$. An application essentially removes a lambda abstraction
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\Rightarrow^{\sigma} A$. An application essentially removes a lambda abstraction
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introduced earlier in the derivation tree. The rule for this inference case is a
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bit more settle, especially because of the usage variables.
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To introduce the term of an application, $\mathsf{App}(f,x)$, it requires to
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give/have a judgement saying that $f$ is a (dependent) function, i.e., $\Gamma
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\vdash f :^{\sigma} (x : ^\pi A) \to B$, for usages variables $\sigma$ and
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\vdash f \overset{\sigma}{:} (x \overset{\pi}{:} A) \to B$, for usages variables $\sigma$ and
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$\pi$. Then, given $\Gamma$, the function $f$ uses $\pi$ times its input,
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mandatory. We therefore need $\sigma\pi$ resources of an input for $f$ if we
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want to apply $f$ $\sigma$ times, as in the conclusion $\Gamma \vdash
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\mathsf{App}(f,x) \Rightarrow^{\sigma} M$.
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\mathsf{App}(f,x) \Rightarrow^{\sigma} A$.
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In summary, the elimination rule is often presented as follows.
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@ -578,6 +659,8 @@ $$\begin{gathered}
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\Gamma \vdash f {\color{blue}\Rightarrow}^{\sigma}(x : ^\pi A) \to B
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\qquad
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\sigma\pi\cdot\Gamma' \vdash x {\color{red}\Leftarrow}^{\sigma\pi} A
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\qquad
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\color{gray}{0 \cdot \Gamma = 0 \cdot \Gamma'}
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}{
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\Gamma + \sigma\pi\cdot\Gamma' \vdash \mathsf{App}(f,x) \,{\color{blue}\Rightarrow^{\sigma}}\, B
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}
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@ -590,12 +673,12 @@ emphasising the usage bussiness.
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1. $$\begin{gathered}
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\rule{App{\Rightarrow_1}}{
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\Gamma \vdash f {\color{blue}\Rightarrow^{\sigma}} (x :^{\pi} A) \to B
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\qquad
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\color{green}{\sigma \cdot \pi = 0}
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\qquad
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\Gamma \vdash f {\color{blue}\Rightarrow^{\sigma}} (x :^{\pi} A) \to B
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\qquad
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0\cdot \Gamma \vdash x {\color{red}\Leftarrow^{0}} A
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\quad
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\qquad
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}{
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\infer{\Gamma}{\mathsf{App}(f,x)}{\sigma}{B}
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}
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@ -605,11 +688,13 @@ $$
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2. $$\begin{gathered}
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\rule{App{\Rightarrow_2}}{
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\infer{\Gamma_1}{f}{\sigma}{(x :^{\pi} A) \to B}
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\quad
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\color{green}{\sigma \cdot \pi \neq 0}
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\quad
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\qquad
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\infer{\Gamma_1}{f}{\sigma}{(x :^{\pi} A) \to B}
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\qquad
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\check{\Gamma_2}{x}{1}{A}
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\qquad
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\color{gray}{0 \cdot \Gamma_1 = 0 \cdot \Gamma_2}
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\quad
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}{
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\infer{\Gamma_1 + \sigma \pi\cdot \Gamma_2}{\mathsf{App}(f,x)}{\sigma}{B}
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@ -642,8 +727,8 @@ infer σ (App f x) = do
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In the rules above, we have the lemma:
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||||
- $1 \cdot \Gamma \vdash x :^1 M$ entails that $\rho \cdot \Gamma \vdash x
|
||||
:^\rho M$ for any usage $\rho$.
|
||||
- $1 \cdot \Gamma \vdash x :^1 A$ entails that $\sigma \cdot \Gamma \vdash x
|
||||
:^\sigma A$ for any usage $\sigma$.
|
||||
|
||||
|
||||
## Tensor type
|
||||
@ -656,23 +741,26 @@ type. In our case, the principal judgemente shows up in the first premise, which
|
||||
is the fact that $M$ is a tensor product type. If we infer that, the types $A$
|
||||
and $B$ become known facts.
|
||||
|
||||
TODO: use notebooks write up
|
||||
|
||||
|
||||
$$\begin{gathered}
|
||||
\rule{TensorElim{\Rightarrow}}{
|
||||
\Gamma_{1} \vdash M \,\overset{\sigma}{\color{blue}\Rightarrow}\,(x\overset{\pi}{:}A)\otimes B
|
||||
\infer{\Gamma_{1}}{M}{\sigma}{(x\overset{\pi}{:}A)\otimes B}
|
||||
\\
|
||||
0\cdot \Gamma_{1},z\overset{0}{:}(x\overset{\pi}{:}A)\otimes B \vdash C \,\overset{0}{\color{red}\Leftarrow}\,\mathcal{U}
|
||||
\qquad
|
||||
\Gamma_{2},x\overset{\sigma\pi}{:}A, y\overset{\sigma}{:}B\vdash N \overset{\sigma}{\color{red}{\Leftarrow}}C[(x,y)/z]\qquad
|
||||
\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 \mathrm{let}\,z@(x,y)=M\,\,\mathrm{in}\,\,N :C \overset{\sigma}{\color{blue}\Rightarrow}\, C[M/x]
|
||||
\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}
|
||||
%
|
||||
$$
|
||||
|
||||
TODO: I need the syntax "let" as we need pattern-matching to make sure one uses all the components in the pair. Exactly the same situation occurs in linear logic, which actually inspires this rule.
|
||||
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. Then, such a mechanism needs to be introduced, using the keyword $\mathsf{let}\mbox{-}\mathsf{in}$.
|
||||
|
||||
## Sum type elim
|
||||
|
||||
|
Loading…
Reference in New Issue
Block a user