Crate minitt

Reading the README is recommended before reading the documentation.

Tutorial

Here's a brief summary of the complete type-checking process.

Since this implementation is actually a dialect or, a variation of the original one, I use minitt to represent this implementation and Mini-TT for the original one.

Here's a "feature list" (only language features that affect type-checking are listed):

First, Mini-TT supports:

• Pi/Sigma types
• First-class sum types and case-split
• Recursion
• Mutual recursion

Mini-TT does not support (while you may expect it to support):

• Dependent pattern matching (with unification)
• Meta variables, say, implicit arguments

Mini-TT does not, but minitt does support:

• Infer types of expressions that appears deeply inside an expression
• Constant expressions with type signature completely inferred
• Universe levels and its subtyping
  • Notice: 1 is of level 0, Type0 is of level 1
• First-class sum types, its merging operation (so you can merge two sums) and its subtyping (like Sum { A } is a subtype of Sum { A | B })

Version 0.1.8 of minitt is basically a vanilla Mini-TT, several extensions are introduced in later versions.

For those who want to have a try on minitt: Please do notice that function application in minitt is right-associative, which is very-very (very-very-very-very) anti-intuitive. This is because the parser is implemented primarily for debugging the type-checker, it's not for general-purpose programming. If you want to write some real code, I recommend Voile.

Syntax Trees

Mini-TT has three syntax trees:

• Surface syntax tree, aka concrete syntax tree, representing expressions that cannot be type-checked alone (and this is because of the presence of free variables) or simply not type-checked yet
• Abstract syntax tree, aka values or terms, representing expressions that are already type-checked. This implies "no free variables"
  • Values might be neutral values: these values represents variable bindings that are not free but not reducible, like a parameter, or an expression that cannot be reduced due to the presence of this parameter
  • Values might be closures: surface syntax term + context + parameter bindings
• Normal form syntax tree, aka normal forms. This is the output of the "read back" functions
  • Details are introduced later. Personally, I consider this not necessary and ugly.

Mini-TT has two "environments":

• One typing context (called Gamma in minitt), this is passed around only during type-checking
• One evaluation context (called Telescope in minitt), this is stored in closures (as captured environment)

However, since type-checking produces new closures and need a evaluation context to reduce terms, these two environments are passed together during type-checking.
They're stored together in TCS, which is short for "Type-Checking State".

Type-Checking

Mini-TT supports inferring types of simple expressions like applications, variable references, etc. But not the case for even a bit more complicated structures, like lambdas.

checkD

$$ \textnormal{checkD}\quad \rho,\Gamma\vdash_l D\Rightarrow \Gamma' $$

Each program is a sequence of definitions, each definition comes with a type signature and a body expression. We check the definitions one by one, after checking each definition we add it to the context and check the rest. For recursive definitions, we generate a neutral value before actually checking it.

This part is trivial in Mini-TT, but minitt extended definitions with prefix parameters, which are parameters present before the type signature and the body expression, resulting in a much more complicated implementation.

• Outer wrapper
• Recursive checking
• Non-recursive checking
• Prefix parameters checking

check

$$ \textnormal{check}\quad \rho,\Gamma\vdash_l M\Leftarrow t $$

This is the so-called instance of check, the function name in Mini-TT paper is check.

All definitions in Mini-TT comes along with a type signature, Mini-TT tries to type-check the signature and then try to match the body expression with the signature, using some hard-coded patterns (relevant codes are in check/expr.rs), like if the type is a pi-type and the value is a lambda, then we go on checking their bodies and types with the parameter instantiated as a generated value then recursively check if the instantiated body expression is an instance of the pi-type's return type; if the type is a sum type and the value is a constructor call, then check if the constructor is present in the sum.

If all these hard-coded rules are not applicable, infer the expression type and perform a subtyping check. This rule is an extension. The subtyping check is basically doing some hard-coded comparisons as well.

If it still fails, read back to normal form and do a syntactic comparison with the read-backed expected type signature.

checkI

$$ \textnormal{checkI}\quad \rho,\Gamma\vdash_l M\Rightarrow t $$

Try to infer the type of a given expression.

Mini-TT cannot infer types of lambdas or other complicated expressions like nested function calls, but in minitt this situation has changed with the help of prefix parameters.

checkT

$$ \textnormal{checkT}\quad \rho,\Gamma\vdash_l A $$

Check if an expression is a type expression, returns the type's level because only Value have the method level() (while returning the level is the only way for check_type to preserve this information. If we return a value, we can call .level() on the value).

Use some hard-coded rules and fallback to check(expr, Type).

Possible Extensions

Several extensions can be made apart from the improvements that have nothing to do with the core type theory. I'm listing all the possible extension, disregarding how hard can the implementation be.

• Indexed inductive families
  • Dependent (co)pattern matching
  • Overlapping pattern matching
  • Props
  • Without-K
• Quantitative Type Theory
  • Linear Type System
    • Symmetric-Interaction-Calculus
  • Affine Type System
• First-class cases and sums
  • Record polymorphism
  • Extensible sums
• Cubical Type Theory
  • Already implemented in another Mini-TT dialect: cubicaltt
  • Cartesian Cubical Type Theory
  • De Morgan Cubical Type Theory
  • Higher-Inductive Families
    • First-class?
• Coinduction and Guarded Recursion
• Sized types (implicit?)

Modules

ast	Syntax: term, expression, context.
check	Type checking: everything related to type-checking.
eval	Reduction: eval and eval's friends.
parser	Parser, from text to AST and a bunch of related tools.
pretty	Pretty print utilities.

Macros

tce_unreachable	Records the source code location that the error occurs, just like Agda.Util.Impossible
tcs_borrow	Cannot be an implementation of Clone due to lifetime requirement.