# Questions tagged [typed-lambda-calculus]

The tag has no usage guidance.

74 questions
Filter by
Sorted by
Tagged with
42 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 ...
203 views

### Is there a text that discusses both the “lambda cube” of pure type theories and Martin-Löf's intuitionistic type theories, and compares them?

I am lost in a maze of twisty little type theories, all different. There are a number of works (textbooks and papers) that discuss pure type theories, and specifically the ones constituting the ...
29 views

### Which family of bicartesian closed functors can define the semantics of simply typed lambda calculus with products and sums

Given any bicartesian closed category $\mathbf{C}$, any natural number $n \geq 0$, and any vector $\boldsymbol{A} \in \mathbf{C}^n$ with $n$ objects $A_1, A_2, … A_n \in \mathbf{C}$, how can I define ...
74 views

### What is the type of the lambda term $\lambda a.a(\lambda yt.t)(ya)$?

I was given an exercise that asked me to assign a simple type to the lambda term: $$\lambda a.a(\lambda yt.t)(ya)$$ but I couldn't find one, furthermore, the lambda term seems untypable to me ...
1 vote
81 views

### Intuition behind UTT's internal logic

The "internal logic" of type theory UTT is defined in LF as follows: What's the intuition behind this definition? I can kind of understand the declaration of the the first three constants - ...
324 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 ...
89 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 ...
255 views

### Complexity of convertibility in simply typed λ-calculus with sums

For the simply typed λ-calculus with only the function type →, the complexity of deciding βη-equivalence is well-understood: it's TOWER-complete (as mentioned here). I expect the same should be true ...
1 vote
67 views

144 views

### How to implement the next type inference algorithm?

Here I mean only simple typed Lambda calculus / Combinatory logic. Notation: Combinatory logic terms: $F, X_i, Y_i$. Term application: $(F*X_1)$. Type variables $x_i,y_i$. Type assignment: $X:x_i$. ...
168 views

### Stratification of System Fω

I'm wondering if there's any update on this conjecture listed by Urzyczyn from years and years ago (I don't think that's its first appearance), which I'll restate below. System Fω can be stratified ...
1 vote
148 views

### What is the model of computation that corresponds (in the manner of Curry-Howard) to the deduction rule of resolution?

The Curry-Howard Correspondence is well-documented for the isomorphism which associates the intuitionistic natural deduction proof calculus (logic side) with the type system for the simply typed ...
78 views

### Question about "Free-ness" of Free SCWF

In Category with Family by Castellan et al., they introduce the concept of Free SCWF as correspondence of STLC with base type. Seemingly, they define Free B-SCWF as the synonym of initial B-SCWF. My ...
1 vote
60 views

### Question in relating STLC and Free CCC

In Lambek's Intro to Higher Order Cat Logic, Chapter 1 Section 4 introduces the free construction (upon graph) My question is, if I want to have STLC + (fake/incomplete) boolean type, how do I have ...
88 views

### Can you always throw away the types when evaluating lambda expressions?

As I understand it, in the simply typed lambda calculus you can type-check your terms, then throw away all the types and do the evaluation exactly as if it was the untyped lambda calculus. Is that ...
186 views

### $\lambda$-definability and structure preserved by homomorphisms

I imagine there are some standard results that bear on this, but I'm having trouble finding a proof or refutation of it. Some prelimary definitions. A Henkin structure $A = (A^\cdot, ⟦\cdot⟧_A)$ for ...
201 views

### Constructive Strong Normalization of the Extended Calculus of Constructions

The extended calculus of constructions (ECC) is basically the calculus of constructions with cumulative universes. I use the definition which Zhaohui Luo used in his PhD theses which contained a proof ...
502 views

### Are there strongly normalizing lambda terms that cannot be given a System F type?

I know that all well-typed System F terms are strongly normalizing, but is the converse true as well? In other words, does System F typeability precisely characterize program termination? (And if so, ...
161 views

### Set-theoretic encoding of functions in type theory

Functions usually get encoded in set theory as follows. A function $A\to B$ is a subset $f\subset A\times B$ such that $\pi_1:f\to A$ is a bijection. In type theory to give a function $A\to B$ is to ... 172 views

### Dependent eliminator for empty type in intensional Martin-Löf type theory

In calculus of inductive constructions you can just say that the empty type is the type with no constructors and it automatically builds the dependent eliminator. But let's say I'm setting up ... 102 views

A known benefit of the HM type system is that you can usually infer a term's most general type with no user-provided type annotations. For example, if my theory contains the standard axiom: $$\forall ... 1 vote 2 answers 186 views ### Explicit type system with infinite non-cumulative universe hierarchy Is there an open-source proof assistant or at least an explicit set of rules written down somewhere for a type system with an infinite non-cumulative universe hierarchy and unique typing? I want to ... 2 votes 2 answers 150 views ### \mathbb{N} in intensional MLTT with judgmentally commutative + and \times Is there a way to implement natural numbers in intensional Martin-Löf type theory so that addition and multiplication is judgmentally commutative? 4 votes 1 answer 219 views ### Model of MLTT with \eta rule where function extensionality fails Consider intensional Martin-Löf type theory with judgmental \eta rule for dependent product types. Is there a model of it where function extensionality fails? 2 votes 1 answer 88 views ### Normal term of double negation of W-type Consider the intensional Martin-Löf type theory without axiom of choice or the law of excluded middle. Let A:U_0 be a type and B:A\to U_0 be a function such that \Sigma_{a:A}(B(a)\to 0)is ... 2 votes 1 answer 107 views ### Typing inference as a map on abstract syntax trees Is there a reference that explains typing inference for Martin-Löf type theory as a computable map from abstract syntax trees of terms to abstract syntax trees of types? I don't want to identify non-... 5 votes 1 answer 215 views ### Is the Mendler-encoding in System-F adequate? In the paper "Efficiency of Lambda-Encodings in Total Type Theory" it is mentioned that the Church-encoding is adequate and the Parigot encoding is not adequate. This means that any ... 4 votes 1 answer 472 views ### Complexity of type inference in the simply typed lambda calculus A similar question was answered here: Is simply typed lambda calculus equivalent to primitive recursive functions What I conclude from the answers is that the complexity is that of the extended ... 7 votes 0 answers 233 views ### Category theory lambda cube? If simply typed lambda calculus corresponds to cartesian closed categories, what types of categories do other calculi in the lambda cube correspond to? https://en.m.wikipedia.org/wiki/Lambda_cube 10 votes 2 answers 569 views ### Is there an efficient beta-equivalence algorithm? Is there an efficient algorithm to determine if two terms are beta-equivalent? Specifically, I am curious about simply-typed-lambda-calculus, so you can assume both terms are strongly normalizing. I ... -2 votes 1 answer 117 views ### Beta reduction and vacuous lambda abstraction [closed] Suppose we have the following typed lambda term (where s does not occur in E (which is of type s \to p) and s and s' have the same type), and want to apply \beta-reduction: (\lambda s. E)\, ... 6 votes 1 answer 156 views ### What arithmetical theorems can plain \lambda \Pi reason about? I've read that System F cannot state or prove theorems of the First Order Theory of Arithmetic. I assume this is because we lack dependent types, so we cannot explicitly express \forall n:\mathbb{N}.... 6 votes 1 answer 229 views ### Decidability of rank-k polymorphism vs. System F There's a paper by Kfoury from 1992, "Type Reconstruction in Finite Rank Fragments of the Second-Order \lambda-Calculus", that proves that type inference for Curry-style rank-k polymorphic lambda ... 3 votes 1 answer 194 views ### Type of induction principle for fixpoint types To the Calculus of Constructions we could add a general fixpoint type constructor (accepting inconsistencies or assuming F is a ... 3 votes 0 answers 165 views ### Type System Of \lambda\mu-Calculus reading this paper on CPS-tranformation from the \lambda\mu-calculus, I'm a bit confused about the type system presented: Why second-order formulas in the types? Is this according to the Curry-... 5 votes 2 answers 258 views ### What's the point of stack judgement in CBPV? Call-by-push-value (CBPV) introduces two main families of types, values and computations, and their corresponding judgements. However, in some extensions/variants/adaptation of CBPV, there is a third ... 3 votes 0 answers 393 views ### Is System-F with higher-kinded newtypes equivalent in computational power to System-F omega? If we have System-F with higher-kinded types and newtypes, then we can express everything (I think) of System-F omega, except we have to manually (un)pack. For example: ... 6 votes 1 answer 317 views ### Fixed set of type constructors to simulate all intensional inductive families? I'm wondering, are there small dependent calculi that can simulate a language with inductive families (that is, has a type isomorphic to each inductive family, at least as powerful of induction ... 6 votes 1 answer 232 views ### Structural equality of Pi Types with heterogeneous equality? I'm trying to implement a proof of the following type: ... 7 votes 2 answers 393 views ### Termination checking for Scott-encodings in System F with positive-recursive types Is there any research on termination analysis on Scott-encodings in System F with positive-recursive types. All papers I have found use languages with constructors and case analysis (for example ... 11 votes 1 answer 461 views ### Understanding the Proof of Strong Normalization of the Calculus of Constructions I have difficulties in understanding the proof of strong normalization for the calculus of constructions. I try to follow the proof in the paper of Herman Geuvers "A short and flexible proof of Strong ... 6 votes 2 answers 295 views ### Typing of substitution in a bidirectional type system In most typed lambda calculi, we have the following lemma: If \Gamma \vdash t_1 : \tau_1 and \Gamma, x : \tau_1, \Delta \vdash t_2 : \tau_2 then \Gamma,\Delta[t_1/x] \vdash t_2[t_1/x] : \tau_2[... 12 votes 1 answer 941 views ### Proof techniques for showing that dependent type checking is decidable I'm in a situation where I need to show that typechecking is decidable for a dependently-typed calculus I'm working on. So far, I've been able to prove that the system is strongly normalizing, and ... 9 votes 1 answer 467 views ### What's the difference between Moggi's computational metalanguage and Moggi's lambda calculus? This is a reference confusion. Sometimes I see people use the term "Moggi's computational metalanguage" to refer to the calculus presented by Moggi, and sometimes to "Moggi's computational lambda ... 3 votes 2 answers 319 views ### Moggi's computational metalanguage In Notions of Computation and Monads Moggi models the notion of a computation of type A, TA, using a monad T. Among other things this ensures the T\eta rule:$$\frac{x: A \vdash a:TB}{x:A \...
We work in the Church-style simply typed lambda calculus. All terms shall be considered in long normal form. Any term of type $A_1\rightarrow A_2\ldots\rightarrow A_n \rightarrow 0$ is of the form \$\...