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2 votes
0 answers
87 views

Variable opening in locally-nameless representation

Although similar to a previously unanswered question, my query focuses on a different aspect of normalization. I'm trying to adjust the proof of strong normalization of STLC, given in Jeremy Avigad's ...
5 votes
2 answers
348 views

Lambda-calculus: Beta-equivalent terms have the same type

In the simply-typed lambda calculus, how do you prove that: If two terms are beta-equivalent, then they have the same type? My guess is that I should use the subject reduction, and maybe the ...
1 vote
0 answers
76 views

Is it possible to define beta reduction for PHOAS?

I'm using Parametric Higher-Order Abstract Syntax (PHOAS) as a representation for untyped lambda calculus in OCaml: ...
6 votes
1 answer
656 views

Strong normalization property of CoC inside CoC

Wikipedia says that The CoC is strongly normalizing, although, by Gödel's incompleteness theorem, it is impossible to prove this property within the CoC since it implies inconsistency. Why is ...
0 votes
0 answers
91 views

Simple Lambda Calculus Question

For any 2 strongly normalizing terms in the simply typed Lambda Calculus, s and t, is st also strongly normalizing? And why? I'm a bit confused as this is used in a proof regarding strong ...
7 votes
1 answer
396 views

The precise definition of Normalization By Evaluation?

The Wikipedia article suggests that NbE is a technique for obtaining "the normal form of terms" by translating the object language into abstractions of the meta (host) language: The ...
7 votes
2 answers
436 views

Is case analysis on normal forms of lambda terms sufficient to prove parametricity results?

There are many closed terms of a given type. For instance, both of these terms: $$ \lambda x . x $$ $$ \lambda x . (\lambda y . y) x $$ have a type of a polymorphic identity function: $$ \forall X ....
1 vote
1 answer
196 views

Defining normalization with respect to judgmental equality instead of reduction

In type theory with a type $\mathbb{N}$ of natural numbers (or some other base type such as booleans) and judgmental equality instead of reductions, canonicity is a meta-theoretical statement claiming ...
5 votes
1 answer
90 views

Can a normal form term be extensionally equivalent to a term with no WHNF?

For convenience I'm using using the combinators SKIBCMTV I notice that it's possible to have a normal-form term extensionally equivalent to a term which has no normal form: ...
2 votes
0 answers
101 views

MLTT/MiniTT: why do normal forms of sum types carry environments?

I am learning how to implement MiniTT: a simple type theoretic language, which is a dependently typed language with sum types, mutual recursive/inductive definitions and a universe of small types. A ...
4 votes
1 answer
131 views

NBE for MiniTT: Why is labelled sum eliminator both a normal form and a neutral value?

I am studying A simple type-theoretic language: MiniTT, which introduces a dependently typed language with The language contains data types, mutual recursive/inductive definitions and a universe of ...
3 votes
1 answer
187 views

References on implementing universe levels over MLTT?

I've followed the tutorial on Checking Dependent Types with Normalization by Evaluation: A Tutorial by David Christiansen, where we implement a small dependently typed kernel with the axiom ...
3 votes
2 answers
538 views

What technique is used to implement type checking for CoC?

I am studying David Christiansen's tutorial on implementing a dependently typed language, where it says: Typed normalization by evaluation is far from the only way to implement conversion checking ...
3 votes
1 answer
242 views

Infinite $\beta \eta$-reduction sequence implies infinite $\beta$-reduction sequence

In Sorensen and Urzyczyn's book there is a lemma (1.3.11) which I am having a hard time proving. 1.3.11 Lemma: If there is an infinite $\beta \eta$-reduction sequence starting with a term $M$ then ...
11 votes
3 answers
320 views

Calculus of Constructions: compress expression to its smallest form

I'm aware that the Calculus of Constructions is strongly normalizing, meaning every expression has a normal for that cannot be beta,eta-reduced further. So in fact this is the most efficient ...
6 votes
0 answers
175 views

Locally-nameless representation: normal order & opening with a bound variable

This question concerns the representation used in Arthur Charguéraud's paper “The locally nameless representation” and is somehow a follow-up on this question, where it is asked about the ...
5 votes
1 answer
257 views

What's the difference between proving weak normalization and implementing evaluator?

Implementing an normalization (cut elimination) procedure for a type system A in a language with a total type system B, automatically proves cut elimination for type system A since the implementation ...
3 votes
1 answer
226 views

Is there a 'very fast growing' hierarchy that would capture System F?

Particular ordinals in slow-growing and fast growing hierarchies can capture the expressiveness of many predicative type systems. Is there a hierarchy of function that could possibly capture ...
2 votes
1 answer
518 views

Is there a formalization of normalization of impredicative system F?

In particular Agda seems not strong enough to prove that. Is the predicative Calculus of Inductive Constructions universes (Coq without Prop) sufficient? How about with the impredicative Prop?
4 votes
1 answer
454 views

Converting Kuroda normal form rules to the Penttonen normal form

Let us say we have some abstract context-sensitive grammar in the Kuroda normal form, which is where all production rules are of the form: $AB\rightarrow CD$ or $A\rightarrow BC$ or $A\rightarrow B$...
1 vote
2 answers
241 views

Proof that the calculus of constructions extended with recursive types isn't strongly normalizing?

What is the proof that the calculus of constructions, extended with recursive types (i.e., Fix at the type-level) isn't strongly normalizing?
8 votes
1 answer
489 views

What is the formal definitions of the reduction related to the "call/cc" (call with the current continuation) operator?

In lambda calculus or in combinatory logic we formally define reduction/expansion rules for terms (and in their typed variants reductions must preserve the type). Then we can talk about properties of ...
18 votes
2 answers
790 views

Is it possible to decide $\beta$-equivalence within System F (or another normalizing typed λ-calculus)?

I know that's impossible to decide $\beta$-equivalence for untyped lambda calculus. Quoting Barendregt, H. P. The Lambda Calculus: Its Syntax and Semantics. North Holland, Amsterdam (1984).: If A ...
3 votes
0 answers
138 views

Expansion normal forms of confluent term rewriting systems

Suppose one has two rewrite rules $\to^\eta,\to^\beta$, both of which are confluent and such that $\to^A := \to^{(\eta \cup \beta)}$ is also confluent. Define a $\beta$-normal form relative to $\eta$ ...
3 votes
1 answer
191 views

Explanation of definition of normalizing: 9.1.12 in Terese "Term Rewriting Systems"?

A strategy for a rewriting system is a sub-rewriting system with the same objects and same normal forms. Definition (from Terese "Term Rewriting Systems"). Let N be a superset of the normal forms of ...
16 votes
3 answers
1k views

Can we prove weak normalization for System F by induction on a transfinite ordinal

Weak normalization for the simple typed lambda calculus can be proved (Turing) by induction on $\omega^2$. An extended lambda calculus with recursors on natural numbers (Gentzen) has a weak ...
4 votes
2 answers
958 views

Canonical forms for relational algebra expressions

I'm looking for whatever work may exist, or thoughts people have, on the question of whether/to what extent there exist(s) one or more canonical form(s) to which relational algebra expressions may be ...
2 votes
2 answers
272 views

formalizing a statement about the expressive power of programming languages wrt divergence

In the Coq'Art book the authors mention in passing that any language that can calculate all computable functions must also be able to express diverging computations. Or in other words, there can be no ...
16 votes
1 answer
584 views

Are innermost reductions perpetual in untyped λ-calculus?

(I have already asked this at MathOverflow, but got no answers there.) Background In the untyped lambda calculus, a term may contain many redexes, and different choices about which one to reduce may ...