Homotopy Type System

Type theory with two equalities

SYNOPSIS

While we are on our road to Lean-like tactic language, currently we are at the stage of regular cubical HTS type checker with CHM-style [5] primitives. You may try it at Github: groupoid/anders or install through OPAM:

$ opam install anders $ anders Anders theorem prover [MLTT][CCHM][HTS] version 0.7.2 invocation = anders | anders list list = [] | command list primitive = zero | one | interval command = check <filename> | lex <filename> | parse <filename> | prim primitive <name> | cubicaltt <filename> | girard | trace | verbose | repl | help

Anders is fast, idiomatic and educational. We carefully draw the favourite Lean-compatible syntax to fit 130 LOC in Menhir, the MLTT core is 500 LOC and CCHM [4] pretypes is another 500 LOC. It compiles under 2 seconds and checks all the base library under 1 second [Haswell, 2013].

SYNTAX

The syntax resembles original syntax of the reference CCHM type checker cubicaltt, is slightly compatible with Lean syntax and contains the full set of Cubical Agda [10] primitives.

Here is given the mathematical pseudo-code notation of the language expressions that come immediately after parsing. The core syntax definition of HTS language $E$ corresponds to exp type defined in expr.ml OCaml module:

$$ E := cosmos\ |\ var\ |\ MLTT\ |\ CCHM\ |\ HIT \\ HIT := \mathbf{inductive}\ E\ E\ |\ \mathbf{ctor}\ name\ E\ |\ \mathbf{match}\ E\ E \\ CCHM := path\ |\ I\ |\ part\ |\ sub\ |\ kan\ |\ glue \\ MLTT := pi\ |\ sigma\ |\ id \\ $$ $$ cosmos := \mathbf{U}_j \ |\ \mathbf{V}_k \\ var := \mathbf{var}\ name\ |\ \mathbf{hole} \\ pi := \Pi\ name\ E\ E\ |\ \lambda\ name\ E\ E\ |\ E\ E \\ sigma := \Sigma\ name\ E\ E\ |\ (E,E)\ |\ E.1\ |\ E.2 \\ id := \mathbf{Id}\ E\ |\ \mathbf{ref}\ E\ |\ \mathbf{idJ}\ E \\ path := \mathbf{Path}\ E\ |\ E^i\ |\ E\ @\ E \\ I := \mathbf{I}\ |\ 0\ |\ 1\ |\ E\ \meet\ E\ |\ E\ \join\ E\ |\ \neg E \\ part := \mathbf{Partial}\ E\ E\ |\ \mathbf{[}\ (E=I) \rightarrow E, ...\ \mathbf{]} \\ sub := \mathbf{inc}\ E\ |\ \mathbf{ouc}\ E\ |\ E\ \mathbf{[}\ I\ \map\ E\ \mathbf{]} \\ kan := \mathbf{transp}\ E\ E\ |\ \mathbf{hcomp}\ E \\ glue := \mathbf{Glue}\ E\ |\ \mathbf{glue}\ E\ |\ \mathbf{unglue}\ E\ E $$ $$ $$

Further Menhir BNF notation will be used to describe the top-level language parser as type checker is written in OCaml.

Keywords. The words of a top-level language (file or repl) consist of keywords or identifiers. The keywords are following: module, where, import, option, def, axiom, postulate, theorem, (, ), [, ], <, >, /, .1, .2, Π, Σ, ,, λ, ×, , :, :=, , U, V, , , -, +, @, PathP, transp, hcomp, zero, one, Partial, inc, ouc, interval, inductive, Glue, glue, unglue.

Indentifiers. Identifiers support UTF-8. Indentifiers couldn't start with :, -, . Sample identifiers:

¬-of-∨, 1=1, is-?, =, $^~]!005x, ∞, x→Nat, f'

Modules. Modules represent files with declarations. More accurate, BNF notation of module consists of imports, options and declarations.

Imports. The import construction supports file folder structure (without file extensions) by using reserved symbol / for hierarchy walking.

Options. Each option holds bool value. Language supports following options: 1) girard (enables U : U); 2) pre-eval (normalization cache); 3) impredicative (infinite hierarchy with impredicativity rule);

%start <Module.file> file %start <Module.command> repl repl : COLON IDENT exp1 EOF | COLON IDENT EOF | exp0 EOF | EOF file : MODULE IDENT WHERE line* EOF path : IDENT line : | IMPORT path+ | OPTION IDENT IDENT | declarations

In Anders you can enable or disable language core types, adjust syntaxes or tune inner variables of the type checker. Here is the example how to setup minimal core able to prove internalization of MLTT-73 variation (Path instead of Id and no inductive types, see base library):

module MLTT-Intern where option HIT false option CCHM false option MLTT true option Path true option Interval true option Transport true import lib/mltt

In order to turn HIT into ordinary CiC calculus you may say:

module CiC where option CCHM false option MLTT true option HIT true import lib/list

Declarations. Language supports following top level declarations: 1) axiom (non-computable declaration that breakes normalization); 2) postulate (alternative or inverted axiom that can preserve consistency); 3) definition (almost any explicit term or type in type theory); 5) lemma (helper in big game). 4) theorem (something valuable or complex enough).

ident : IRREF | IDENT vars : ident+ lense : LPARENS vars COLON exp1 RPARENS telescope : lense telescope params : telescope | [] declarations: | DEF IDENT params DEFEQ exp1 | DEF IDENT params COLON exp1 DEFEQ exp1 | AXIOM IDENT params COLON exp1

Sample declarations. For example, signature isProp (A : U) of type U could be defined as normalization-blocking axiom without proof-term or by providing proof-term as definition.

axiom isProp (A : U) : U def isSet (A : U) : U := Π (a b : A) (x y : Path A a b), Path (Path A a b) x y

In this example (A : U), (a b : A) and (x y : Path A a b) are called telescopes. Each telescope consists of a series of lenses or empty. Each lense provides a set of variables of the same type. Telescope defines parameters of a declaration. Types in a telescope, type of a declaration and a proof-terms are a language expressions exp1.

Expressions. All atomic language expressions are grouped by four categories: exp0 (pair constructions), exp1 (non neutral constructions), exp2 (path and pi applcations), exp3 (neutral constructions).

face : LPARENS IDENT IDENT IDENT RPARENS partial : face+ ARROW exp1
exp0 : | exp1 COMMA exp0 | exp1
exp1: | LAM telescope COMMA exp1 | PI telescope COMMA exp1 | SIGMA telescope COMMA exp1 | LSQ IRREF ARROW exp1 RSQ | LSQ separated_list(COMMA, partial) RSQ | LT vars GT exp1 | exp2 ARROW exp1 | exp2 PROD exp1 | exp2

The LR parsers demand to define exp1 as expressions that cannot be used (without a parens enclosure) as a right part of left-associative application for both Path and Pi lambdas.

exp2 : | exp2 exp3 | exp2 APPFORMULA exp3 | exp3

Universe indecies Uj (inner fibrant), Vk (outer pretypes) and S (outer strict omega) are using unicode subscript letters that are already processed in lexer.

exp3: | HOLE | PRE | KAN | exp3 FST | exp3 SND | NEGATE exp3 | exp3 AND exp3 | exp3 OR exp3 | ID exp3 | REF exp3 | IDJ exp3 | INC exp3 | OUC exp3 | PATHP exp3 | TRANSP exp3 exp3 | HCOMP exp3 | PARTIAL exp3 | IDENT LSQ exp0 MAP exp0 RSQ | LPARENS exp0 RPARENS | IDENT | LPARENS exp0 RPARENS LSQ exp0 MAP exp0 RSQ

SEMANTICS

The idea is to have a unified layered type checker, so you can disbale/enable any MLTT-style inference, assign types to universes and enable/disable hierachies. This will be done by providing linking API for pluggable presheaf modules. We selected 5 levels of type checker awareness from universes and pure type systems up to synthetic language of homotopy type theory. Each layer corresponds to its presheaves with separate configuration for universe hierarchies.

inductive lang : U | UNI: cosmos → lang | PI: pure lang → lang | SIGMA: total lang → lang | ID: uip lang → lang | PATH: homotopy lang → lang | GLUE: gluening lang → lang | HIT: hit lang → lang

We want to mention here with homage to its authors all categorical models of dependent type theory: Comprehension Categories (Grothendieck, Jacobs), LCCC (Seely), D-Categories and CwA (Cartmell), CwF (Dybjer), C-Systems (Voevodsky), Natural Models (Awodey). While we can build some transports between them, we leave this excercise for our mathematical components library.

We will use here the Coquand's notation for Presheaf Type Theories in terms of restriction maps.


Universe Hierarchies

Language supports Agda-style hierarchy of universes: fibrant (U), interval pretypes (V) and strict omega with explicit level manipulation. All universes are bounded with preorder $$ Fibrant_j \prec Pretypes_k \prec Strict_l, $$

in which $j,k,l$ are bounded with equation: $$ j < k < l. $$

Large elimination to upper universes is prohibited. This is extendable to Agda model:

inductive cosmos : U | prop: nat → cosmos | fibrant: nat → cosmos | pretypes: nat → cosmos | strict: nat → cosmos | omega: cosmos | lock: cosmos

Dependent Types

Definition (Type). A type is interpreted as a presheaf $A$, a family of sets $A_I$ with restriction maps $u \mapsto u\ f, A_I \rightarrow A_J$ for $f: J\rightarrow I$. A dependent type B on A is interpreted by a presheaf on category of elements of $A$: the objects are pairs $(I,u)$ with $u : A_I$ and morphisms $f: (J,v) \rightarrow (I,u)$ are maps $f : J \rightarrow$ such that $v = u\ f$. A dependent type B is thus given by a family of sets $B(I,u)$ and restriction maps $B(I,u) \rightarrow B(J,u\ f)$.

We think of $A$ as a type and $B$ as a family of presheves $B(x)$ varying $x:A$. The operation $\Pi(x:A)B(x)$ generalizes the semantics of implication in a Kripke model.

Defintion (Pi). An element $w:[\Pi(x:A)B(x)](I)$ is a familyfunctions $w_f : \Pi(u:A(J))B(J,u)$ for $f : J \rightarrow I$ such$(w_f u)g=w_{f\ g}(u\ g)$ when $u:A(J)$ and $g:K\rightarrow J$.

inductive pure (lang: U) : U | var: name → nat → pure lang | pi: name → nat → lang → lang → pure lang | lambda: name → nat → lang → lang → pure lang | app: lang → lang → pure lang

Defintion (Sigma). The set $\Sigma(x:A)B(x)$ is the setpairs $(u,v)$ when $u:A(I),v:B(I,u)$ and restriction map $(u,v)\ f=(u\ f,v\ f)$.

inductive total (lang: U) : U | sigma: name → lang → lang → total lang | pair: lang → lang → total lang | fst: lang → total lang | snd: lang → total lang

The preseaf configuration with only Pi and Sigma is called MLTT-72.

Path Equality

The fundamental development of equality inside MLTT provers led us to the notion of ∞-groupoid as spaces. In this way Path identity type appeared in the core of type checker along with De Morgan algebra on built-in interval type.

inductive homotopy (lang: U) : U | PathP: lang → lang → lang → homotopy lang | plam: name → lang → lang → homotopy lang | papp: lang → lang → homotopy lang | I: homotopy lang | zero: homotopy lang | one: homotopy lang | meet: lang → lang → homotopy lang | join: lang → lang → homotopy lang | neg: lang → homotopy lang | system: lang → homotopy lang | Partial: lang → homotopy lang | transp: lang → lang → homotopy lang | hcomp: lang → homotopy lang | Sub: lang → homotopy lang | inc: lang → homotopy lang | ouc: lang → homotopy lang

Definition (Cubical Presheaf $\mathbb{I}$). The identity types modeled with another presheaf, the presheaf on Lawvere category of distributive lattices (theory of De Morgan algebras) denoted with $\Box$ — $\mathbb{I} : \Box^{op} \rightarrow \mathrm{Set}$.

Properties of $\mathbb{I}$. The presheaf $\mathbb{I}$: i) has to distinct global elements $0$ and $1$ (B$_1$); ii) $\mathbb{I}$(I) has a decidable equality for each $I$ (B$_2$); iii) $\mathbb{I}$ is tiny so the path functor $X \mapsto X^\mathbb{I}$ has right adjoint (B$_3$).; iv) $\mathbb{I}$ has meet and join (connections).

Interval Pretypes. While having pretypes universe V with interval and associated De Morgan algebra (∧, ∨, -, 0, 1, I) is enough to perform DNF normalization and proving some basic statements about path, including: contractability of singletons, homotopy transport, congruence, functional extensionality; it is not enough for proving β rule for Path type or path composition.

Generalized Transport. Generalized transport transp adresses first problem of deriving the computational β rule for Path types:

theorem Path-β (A : U) (a : A) (C : D A) (d: C a a (refl A a)) : Equ (C a a (refl A a)) d (J A a C d a (refl A a)) := λ (A : U), λ (a : A), λ (C : Π (x : A), Π (y : A), PathP (<_> A) x y → U), λ (d : C a a (<_> a)), <j> transp (<_> C a a (<_> a)) -j d

Transport is defined on fibrant types (only) and type checker should cover all the cases. Note that transpⁱ (Pathʲ A v w) φ u₀ case is relying on comp operation which depends on hcomp primitive.

transpⁱ N φ u₀ = u₀ transpⁱ U φ A = A transpⁱ (Π (x : A), B) φ u₀ v = transpⁱ B(x/w) φ (u₀ w(i/0)) transpⁱ (Σ (x : A), B) φ u₀ = (transpⁱ A φ (u₀.1),transpⁱ B(x/v) φ(u₀.2)) transpⁱ (Pathʲ A v w) φ u₀ = 〈j〉compⁱ A [φ ↦ u₀ j, (j=0) ↦ v, (j=1) ↦ w] (u₀ j) transpⁱ (Glue [φ ↦ (T,w)] A) ψ u₀ = glue [φ(i/1) ↦ t′₁] a′₁ : B(i/1)
w = transpFill⁻ⁱ A φ v, v : A(i/1) v = transpFillⁱ A φ u₀.1 u : A(j/0), v : A(j/1) transp⁻ⁱ A φ u = (transpⁱ A(i/1−i) φ u)(i/1−i) : A(i/0) transpFillⁱ A φ u₀ = transpʲ A(i/i∧j) (φ∨(i=0)) u₀ : A

Partial Elements. In order to explicitly define hcomp we need to specify n-cubes where some faces are missing. Partial primitives isOne, 1=1 and UIP on pretypes are derivable in Anders due to landing strict equality Id in V universe. The idea is that (Partial A r) is the type of cubes in A that are only defined when IsOne r holds. (Partial A r) is a special version of the function space IsOne r → A with a more extensional equality: two of its elements are considered judgmentally equal if they represent the same subcube of A. They are equal whenever they reduce to equal terms for all the possible assignment of variables that make r equal to 1.

def Partial′ (A : U) (i : I) := Partial A i def isOne : I -> V := Id I 1 def 1=1 : isOne 1 := ref 1 def UIP (A : V) (a b : A) (p q : Id A a b) : Id (Id A a b) p q := ref p

Cubical Subtypes. For (A : U) (i : I) (Partial A i) we can define subtype A [ i ↦ u ]. A term of this type is a term of type A that is definitionally equal to u when (IsOne i) is satisfied.

def sub′ (A : U) (i : I) (u : Partial A i) : V := A [i ↦ u ] def inc′ (A : U) (i : I) (a : A) : A [i ↦ [(i = 1) → a]] := inc a def ouc′ (A : U) (i : I) (u : Partial A i) (a : A [i ↦ u]) : A := ouc a

We have forth and back fusion rules ouc (inc v) = v and inc (outc v) = v. Moreover, ouc v will reduce to u 1=1 when i=1.

Homogeneous Composition. Homogeneous composition hcomp is the answer to second problem: with hcomp and transp one can express path composition, groupoid, category of groupoids (groupoid interpretation and internalization in type theory). One of the main roles of homogeneous composition is to be a carrier in [higher] inductive type constructors for calculating of homotopy colimits and direct encoding of CW-complexes (or internal models as gluening of pushout cells).

def comp (A : I → U) (r : I) (u : Π (i : I), Partial (A i) r) (u₀ : (A 0)[r ↦ u 0]) : A 1 := hcomp (A 1) r (λ (i : I), [(φ : r = 1) → transp (<j> A (i ∨ j)) i (u i φ)]) (inc (transp (<i> A i) 0 (ouc u₀)))

The type checker equations for hcomp primitive are following:

hcompⁱ N [φ ↦ 0] 0 = 0 hcompⁱ N [φ ↦ S u] (S u₀) = S (hcompⁱ N [φ ↦ u] u₀) hcompⁱ U [φ ↦ E] A = Glue [φ ↦ (E(i/1), equivⁱ E(i/1−i))] A hcompⁱ (Π (x : A), B) [φ ↦ u] u₀ v = hcompⁱ B(x/v) [φ ↦ u v] (u₀ v) hcompⁱ (Σ (x : A), B) [φ ↦ u] u₀ = (v(i/1), compⁱ B(x/v) [φ ↦ u.2] u₀.2) hcompⁱ (Pathʲ A v w) [φ ↦ u] u₀ = 〈j〉 hcompⁱ A [ φ ↦ u j, (j = 0) ↦ v, (j = 1) ↦ w ] (u₀ j) hcompⁱ (Glue [φ ↦ (T,w)] A) [ψ ↦ u] u₀ = glue [φ ↦ t₁] a₁ = glue [φ ↦ u(i/1)] (unglue u(i/1)) = u(i/1) : Glue [φ ↦ (T,w)] A
hfillⁱ A [φ ↦ u] u₀ = hcompʲ A [φ ↦ u(i/i∧j), (i=0) ↦ u₀] u₀ : A v = hfillⁱ A [φ ↦ u.1] u₀.1 t₁ = u(i/1) : T a₁ = unglue u(i/1) : A glue [φ ↦ t₁] a1 = t₁ : T

Strict Equality

To avoid conflicts with path equalities which live in fibrant universes strict equalities live in pretypes universes.

inductive strict (lang: U) : U | Id: name → lang → total lang | ref: lang → lang → total lang | idJ: lang → lang → lang → total lang

You may think of (Id A a b) [Coquand] to be the type of pairs $\varphi, \omega$ with $\varphi$ in $\mathbb{F}$ and $\omega$ in (Path A a b) such that $\varphi$ implies $const(\omega)$ where $const(\omega) := \Pi(i:I)[\omega(i)=\omega(0)]$. The reflexivity proof is then an element in (Id A a b).

We use strict equality in HTS for pretypes and partial elements which live in V. The presheaf configuration with Pi, Sigma and Id is called MLTT-73. The presheaf configuration with Pi, Sigma, Id and Path is called HTS.

Glue Types

The main purpose of Glue types is to construct a cube where some faces have been replaced by equivalent types. This is analogous to how hcomp lets us replace some faces of a cube by composing it with other cubes, but for Glue types you can compose with equivalences instead of paths. This implies the univalence principle and it is what lets us transport along paths built out of equivalences.

inductive gluening (lang: U) : U | Glue: lang → lang → lang → gluening lang | glue: lang → lang → gluening lang | unglue: lang → lang → gluening lang

Higher Inductive Types

The further development of induction inside MLTT provers led to the theory of polynomial functors and well-founded trees, known in programming languages as inductive types with inductive and record core primitives.

Inductive types could be encoded in PTS/Cedile using non-recursive representation of Bohm-Berarducci schemes or with categorical impredicative encoding by Steve Awodey. We want to have optional infinite impredicative universe in type checker for such experiments in PTS mode.

Anders supports Lean-compatible inductive definition so we can copy-paste terms between Lean and Anders.

inductive tele (A: U) : U | emp: tele A | tel: name → A → tele A → tele A inductive branch (A: U) : U | br: name → branch A | args: list name → A → branch A inductive label (A: U) : U | lab: name → label A | t: tele A → label A inductive hit (lang: U) | form: name → tele lang → list (label lang) → hit lang | ctor: name → list lang → hit lang | htor: name → list lang → list lang → lang → hit lang | case: name → lang → list (branch lang) → hit lang

The non-well-founded trees or infinite coinductive trees are useful for modeling infinite processes and are part of Milner's Pi-calculus. Coinductive streams could be found in many MLTT base libraries.

ANDERS.PDF

Bibliography


HTS

1) A simple type system with two identity types [Voevodsky]. 2) Two-level type theory and applications [Annenkov, Capriotti, Kraus, Sattler]. 3) Syntax for two-level type theory [Bonacina, Ahrens].

CCHM

4) Cubical Type Theory: a constructive interpretation of the univalence axiom [Cohen, Coquand, Huber, Mörtberg]; 5) On Higher Inductive Types in Cubical Type Theory [Coquand, Huber, Mörtberg]; 6) Canonicity for Cubical Type Theory [Huber]; 7) Canonicity and homotopy canonicity for cubical type theory [Coquand, Huber, Sattler]; 8) Cubical Synthetic Homotopy Theory [Mörtberg, Pujet]; 9) Unifying Cubical Models of Univalent Type Theory [Cavallo, Mörtberg, Swan]; 10) Cubical Agda: A Dependently Typed PL with Univalence and HITs [Vezzosi, Mörtberg, Abel]; 11) A Cubical Type Theory for Higher Inductive Types [Huber]; 12) Gluing for type theory [Kaposi, Huber, Sattler]. 13) Cubical Methods in HoTT/UF [Mörtberg].

MLTT

14) An intuitionistic theory of types [Martin-Löf]; 15) An intuitionistic theory of types: predicative part. [Martin-Löf]; 16) Programming in Martin-Löf’s Type Theory [Nordström, Petersson, Smith]; 17) A simple type-theoretic language: Mini-TT [Coquand, Kinoshita, Nordström, Takeyama];