Pure Type System

OM — language with universal quntifier and infinity number of universes for typechecking and normalization

Article

Universes

The OM language is a higher-order dependently typed lambda calculus, an extension of Coquand's Calculus of Constructions with the predicative/impredicative hierarchy of indexed universes. This extension is motivated avoiding paradoxes in dependent theory. Also there is no fixpoint axiom needed for the definition of infinity term dependance.

U₀ : U₁ : U₂ : U₃ : … U₀ — propositions U₁ — values and sets U₂ — types U₃ — sorts
\begin{equation} \tag{S} \dfrac {o : Nat} {Type_o} \end{equation}

Predicative Universes

All terms obey the A ranking inside the sequence of S universes, and the complexity R of the dependent term is equal to a maximum of the term's complexity and its dependency. The universes system is completely described by the following PTS notation (due to Barendregt):

S (n : nat) = U n A₁ (n m : nat) = U n : U m where m > n — cumulative R₁ (m n : nat) = U m ⟶ U n : U (max m n) — predicative

Note that predicative universes are incompatible with Church lambda term encoding. You can switch predicative vs impredicative uninverses by typecheker parameter.

$$ \begin{equation} \tag{A₁} \dfrac {i: Nat,\ \ \ \ j: Nat,\ \ \ \ i < j} {Type_i : Type_{j}} \end{equation} $$
$$ \begin{equation} \tag{R₁} \dfrac {i : Nat,\ \ \ \ j : Nat} {Type_i \rightarrow Type_{j} : Type_{max(i,j)}} \end{equation} $$

Impredicative Universes

Propositional contractible bottom space is the only available extension to predicative hierarchy that not leads to inconsistency. However there is another option to have infinite impredicative hierarchy.

A₂ (n : nat) = U n : U (n + 1) — non-cumulative R₂ (m n : nat) = U m ⟶ U n : U n — impredicative
$$ \begin{equation} \tag{A₂} \dfrac {i: Nat} {Type_i : Type_{i+1}} \end{equation} $$
$$ \begin{equation} \tag{R₂} \dfrac {i : Nat,\ \ \ \ j : Nat} {Type_i \rightarrow Type_{j} : Type_{j}} \end{equation} $$

Single Axiom Language

This langauge is called one axiom language (or pure) as eliminator and introduction adjoint functors inferred from type formation rule. The only computation rule of Pi type is called beta-reduction.

∀ (x: A) → B x : Type — formation rule λ (x: A) → b : B x — introduction f a : B [a/x] — elimination (λ (x: A) → b) a = b [a/x] : B [a/x] — equation
\begin{equation} \tag{$\Pi$-formation} \dfrac {x:A \vdash B : Type} {\Pi\ (x:A) \rightarrow B : Type} \end{equation}
\begin{equation} \tag{$\lambda$-intro} \dfrac {x:A \vdash b : B} {\lambda\ (x:A) \rightarrow b : \Pi\ (x: A) \rightarrow B } \end{equation}

$$ \begin{equation} \tag{$App$-elimination} \dfrac {f: (\Pi\ (x:A) \rightarrow B)\ \ \ a: A} {f\ a : B\ [a/x]} \end{equation} $$
$$ \begin{equation} \tag{$\beta$-computation} \dfrac {x:A \vdash b: B\ \ \ a:A} {(\lambda\ (x:A) \rightarrow b)\ a = b\ [a/x] : B\ [a/x]} \end{equation} $$

This language could be embedded in itself and used as Logical Framework for the Pi type:

pi (A: U) (B: A -> U) : U = (x: A) -> B x Pi (A: U) : U = (intro: (A -> U) -> U) * (lambda: (B: A -> U) -> pi A B -> intro B) * (app: (B: A -> U) -> intro B -> pi A B) * (applam: (B: A -> U) (f: pi A B) -> (a: A) -> Path (B a) ((app B (lambda B f)) a) (f a)) * (lamapp: (B: A -> U) (p: intro B) -> Path (intro B) (lambda B (\(a:A) -> app B p a)) p) * Unit

Syntax

The terms of OM consist of nat indexed stars, variables, applications, abstractions and universal quantifications. This language is called Calculus of Construction and exists in various syntaxes. OM supports Morte's syntax.

<> = #option I = #identifier U = * < #number > OM = U | I | ( OM ) | λ ( I : OM ) → OM OM OM | OM → OM | ∀ ( I : OM ) → OM

Equivalent tree encoding for parsed terms is following:

Inductive OM := Star: nat → OM | Var: name → OM | App: OM → OM → OM | Lambda: name → OM → OM → OM | Arrow: OM → OM → OM | Pi: name → OM → OM → OM.

Universes

star (:star,N) -> N star _ -> (:error, "*")

Functions

func ((:forall,),(I,O)) -> true func T -> (:error,(:forall,T))

Variables

var N B -> var N B (proplists:is_defined N B) var N B true -> true var N B false -> (:error,("free var",N,proplists:get_keys(B)))

Shift

sh (:var,(N,I)),N,P) when I>=P -> (:var,(N,I+1)) sh ((:forall,(N,0)),(I,O)),N,P) -> ((:forall,(N,0)),sh I N P,sh O N P+1) sh ((:lambda,(N,0)),(I,O)),N,P) -> ((:lambda,(N,0)),sh I N P,sh O N P+1) sh (Q,(L,R),N,P) -> (Q,sh L N P,sh R N P) sh (T,N,P) -> T

Substitution

sub Term Name Value -> sub Term Name Value 0 sub (:arrow, (I,O)) N V L -> (:arrow, sub I N V L,sub O N V L); sub ((:forall,(N,0)),(I,O)) N V L -> ((:forall,(N,0)),sub I N V L,sub O N(sh V N 0)L+1) sub ((:forall,(F,X)),(I,O)) N V L -> ((:forall,(F,X)),sub I N V L,sub O N(sh V F 0)L) sub ((:lambda,(N,0)),(I,O)) N V L -> ((:lambda,(N,0)),sub I N V L,sub O N(sh V N 0)L+1) sub ((:lambda,(F,X)),(I,O)) N V L -> ((:lambda,(F,X)),sub I N V L,sub O N(sh V F 0)L) sub (:app, (F,A)) N V L -> (:app,sub F N V L,sub A N V L) sub (:var, (N,L)) N V L -> V sub (:var, (N,I)) N V L when I>L -> (:var,(N,I-1)) sub T _ _ _ -> T.

Normalization

norm :none -> :none norm :any -> :any norm (:app,(F,A)) -> case norm F of ((:lambda,(N,0)),(I,O)) -> norm (sub O N A) NF -> (:app,(NF,norm A)) end norm (:remote,N) -> cache (norm N []) norm (:arrow, (I,O)) -> ((:forall,("_",0)), (norm I,norm O)) norm ((:forall,(N,0)),(I,O)) -> ((:forall,(N,0)), (norm I,norm O)) norm ((:lambda,(N,0)),(I,O)) -> ((:lambda,(N,0)), (norm I,norm O)) norm T -> T

Definitional Equality

eq ((:forall,("_",0)), X) (:arrow,Y) -> eq X Y eq (:app,(F1,A1)) (:app,(F2,A2)) -> let true = eq F1 F2 in eq A1 A2 eq (:star,N) (:star,N) -> true eq (:var,(N,I)) (:var,(N,I)) -> true eq (:remote,N) (:remote,N) -> true eq ((:farall,(N1,0)),(I1,O1)) ((:forall,(N2,0)),(I2,O2)) -> let true = eq I1 I2 in eq O1 (sub (sh O2 N1 0) N2 (:var,(N1,0)) 0) eq ((:lambda,(N1,0)),(I1,O1)) ((:lambda,(N2,0)),(I2,O2)) -> let true = eq I1 I2 in eq O1 (sub (sh O2 N1 0) N2 (:var,(N1,0)) 0) eq (A,B) -> (:error,(:eq,A,B))

Type Checker

type (:star,N) _ -> (:star,N+1) type (:var,(N,I)) D -> let true = var N D in keyget N D I type (:remote,N) D -> cache type N D type (:arrow,(I,O)) D -> (:star,h(star(type I D)),star(type O D)) type ((:forall,(N,0)),(I,O)) D -> (:star,h(star(type I D)),star(type O [(N,norm I)|D])) type ((:lambda,(N,0)),(I,O)) D -> let star (type I D), NI = norm I in ((:forall,(N,0)),(NI,type O [(N,NI)|D]))) type (:app,(F,A)) D -> let T = type(F,D), true = func T, ((:forall,(N,0)),(I,O)) = T, Q = type A D, true = eq I Q in norm (sub O N A)

Passes

Normalized

Terms in OM language.

$ om show List/Cons λ (A: *) → λ (Head: A) → λ (Tail: ∀ (List: *) → ∀ (Cons: ∀ (Head: A) → ∀ (Tail: List) → List) → ∀ (Nil: List) → List) → λ (List: *) → λ (Cons: ∀ (Head: A) → ∀ (Tail: List) → List) → λ (Nil: List) → Cons Head (Tail List Cons Nil)

Extract Language

The untyped lambda language O is the simplest language used in OM to generate backend programs. This O is used as the output of type erasure.

I = #identifier O = I | ( O ) | O O | λ I -> O
Inductive O := Var: nat → O | App: O → O → O | Lambda: nat → O → O → O

Erased

Terms in untyped lambda pure language.

$ om print fst erase a "#List/Cons" ( λ Head → ( λ Tail → ( λ Cons → ( λ Nil → ((Cons Head) ((Tail Cons) Nil)))))) ok

Erlang extracted code. For other targets you may want to read Code Extract notes.

'Cons'() -> fun (Head) -> fun (Tail) -> fun (Cons) -> fun (Nil) -> ((Cons(Head))((Tail(Cons))(Nil))) end end end end.

Intermediate Language with Dependent Types for Erlang/OTP applications