Syntax and Semantics of Dependent Types
... the empty context. If contains among others the free variable x then we can write [x] to emphasise this and use the notation [M ] for [M=x] in this case. In implementations of type theory many more such conventions are being used and sometimes they are even made part of the ocial syntax. It is ...
... the empty context. If contains among others the free variable x then we can write [x] to emphasise this and use the notation [M ] for [M=x] in this case. In implementations of type theory many more such conventions are being used and sometimes they are even made part of the ocial syntax. It is ...
- Free Documents
... Heyting algebra semantics for intuitionistic logic. The point is that the soundness theorem is nontrivial and therefore some work can be saved when presenting a translation of the syntax as a model construction. ...
... Heyting algebra semantics for intuitionistic logic. The point is that the soundness theorem is nontrivial and therefore some work can be saved when presenting a translation of the syntax as a model construction. ...
Intuitionistic Completeness of First-Order Logic -- mPC Case
... mPC and iPC, and show that it opens a new approach to completeness questions for constructive logics. Here we use the idea to prove constructive completeness theorems with respect to uniform evidence semantics for both minimal and intuitionistic first-order logic. We use the term evidence semantics ...
... mPC and iPC, and show that it opens a new approach to completeness questions for constructive logics. Here we use the idea to prove constructive completeness theorems with respect to uniform evidence semantics for both minimal and intuitionistic first-order logic. We use the term evidence semantics ...
NOTE ON COFIBRATION In this overview I assume
... In this overview I assume, that all the topological spaces are path connected if not stated otherwise. Definition 1. Let X, Y be topological spaces. If a map f : X → Y induces isomorphism on the homotopy grups πn (Y ) ∼ = πn (X) for all n ≥ 0, we call it weak homotopy equivalence (or weak equivalenc ...
... In this overview I assume, that all the topological spaces are path connected if not stated otherwise. Definition 1. Let X, Y be topological spaces. If a map f : X → Y induces isomorphism on the homotopy grups πn (Y ) ∼ = πn (X) for all n ≥ 0, we call it weak homotopy equivalence (or weak equivalenc ...
On Colimits in Various Categories of Manifolds
... understand what’s going on at each step and that in the limit these steps do what is required. For this reason, a homotopy theorist need the freedom to make any construction he wishes, e.g. products, subobjects, quotients, gluing towers of spaces together, dividing out by group actions, moving to fi ...
... understand what’s going on at each step and that in the limit these steps do what is required. For this reason, a homotopy theorist need the freedom to make any construction he wishes, e.g. products, subobjects, quotients, gluing towers of spaces together, dividing out by group actions, moving to fi ...
BASIC ALGEBRAIC TOPOLOGY: THE FUNDAMENTAL GROUP OF
... homotopic via the linear homotopoy defined by ft (s) = (1 − t)f0 (s) + tf1 (s). This means that each f0 (s) travels along the line segments to f1 (s) at a constant speed. Proposition 1.5. Given a topological space X with two endpoints a, b ∈ X, path homotopy is an equivalence relation on the set of ...
... homotopic via the linear homotopoy defined by ft (s) = (1 − t)f0 (s) + tf1 (s). This means that each f0 (s) travels along the line segments to f1 (s) at a constant speed. Proposition 1.5. Given a topological space X with two endpoints a, b ∈ X, path homotopy is an equivalence relation on the set of ...
Homotopy type of symplectomorphism groups of × S Geometry & Topology
... i = 1, 2, 3. It is easy to understand what these classes are in homotopy. For example, x1 is a spherical class, so it represents an element in π1 (Gλ ) and x1 t + tx1 corresponds to the Samelson product [t, x1 ] ∈ π2 (Gλ ). This is given by the map S 2 = S 1 × S 1 /S 1 ∨ S 1 → Gλ induced by the comm ...
... i = 1, 2, 3. It is easy to understand what these classes are in homotopy. For example, x1 is a spherical class, so it represents an element in π1 (Gλ ) and x1 t + tx1 corresponds to the Samelson product [t, x1 ] ∈ π2 (Gλ ). This is given by the map S 2 = S 1 × S 1 /S 1 ∨ S 1 → Gλ induced by the comm ...
pdf
... • The λ-calculus, its variants and their properties (e.g., Church Rosser, Strong Normalisation and Standardisation) [42, 44, 45] • The semantics of intersection type systems with expansion using realisability semantics establishing soundness and completeness [39, 40, 41]. • Type error slicing [31] g ...
... • The λ-calculus, its variants and their properties (e.g., Church Rosser, Strong Normalisation and Standardisation) [42, 44, 45] • The semantics of intersection type systems with expansion using realisability semantics establishing soundness and completeness [39, 40, 41]. • Type error slicing [31] g ...
MoggiMonads.pdf
... was later extended, following a similar methodology, to consider other features of computations like nondeterminism (see [Sha84]), side-effects and continuations (see [FFKD86, FF89]). The calculi based only on operational considerations, like the λv -calculus, are sound and complete w.r.t. the opera ...
... was later extended, following a similar methodology, to consider other features of computations like nondeterminism (see [Sha84]), side-effects and continuations (see [FFKD86, FF89]). The calculi based only on operational considerations, like the λv -calculus, are sound and complete w.r.t. the opera ...
LOVELY PAIRS OF MODELS: THE NON FIRST ORDER CASE
... If this holds (the “good” case), then T P is simple as well, and provides an elegant means for the study of certain independence-related properties of T itself, as mentioned above. If this fails (the “bad” case), then the first order theory T P is pretty much useless. This was noticed by Poizat in t ...
... If this holds (the “good” case), then T P is simple as well, and provides an elegant means for the study of certain independence-related properties of T itself, as mentioned above. If this fails (the “bad” case), then the first order theory T P is pretty much useless. This was noticed by Poizat in t ...
Interactive Theorem Proving in Coq and the Curry
... In Coq, type-checking is done with respect to an environment, determined by the declaration and definitions that were executed earlier. A declaration is used to attach a type to an identifier, without giving the value. For example, the declaration of an identifier x with type A is written (x : A). O ...
... In Coq, type-checking is done with respect to an environment, determined by the declaration and definitions that were executed earlier. A declaration is used to attach a type to an identifier, without giving the value. For example, the declaration of an identifier x with type A is written (x : A). O ...
A Calculus for Type Predicates and Type Coercion
... – The type of a term is given by the signature, more precisely, a term’s type is the type declared as return type of the term’s outermost function symbol. The syntax is defined in such a way that a term of a certain type can be used wherever the syntax calls for a term of a supertype. To make it clea ...
... – The type of a term is given by the signature, more precisely, a term’s type is the type declared as return type of the term’s outermost function symbol. The syntax is defined in such a way that a term of a certain type can be used wherever the syntax calls for a term of a supertype. To make it clea ...
Morley`s number of countable models
... each α < ω1 or there is a least α such that they have different Lα -types. Since there is only a countable number of pairs of finite sequences, there must be a δ < ω1 such that if two sequences have the same Lδ -type, then they have the same type for all α < ω1 . Each Lδ -type corresponds to a singl ...
... each α < ω1 or there is a least α such that they have different Lα -types. Since there is only a countable number of pairs of finite sequences, there must be a δ < ω1 such that if two sequences have the same Lδ -type, then they have the same type for all α < ω1 . Each Lδ -type corresponds to a singl ...
About dual cube theorems
... the first cube theorem, here called Axiom 13. Before we state this, we need to give the dual notion of homotopy pull back extension, which we will call ‘homotopy push out coextension’: Definition 12 Any homotopy commutative square as in Definition 6 will be called ‘homotopy push out coextension’ (or ...
... the first cube theorem, here called Axiom 13. Before we state this, we need to give the dual notion of homotopy pull back extension, which we will call ‘homotopy push out coextension’: Definition 12 Any homotopy commutative square as in Definition 6 will be called ‘homotopy push out coextension’ (or ...
ALGEBRAIC TOPOLOGY Contents 1. Preliminaries 1 2. The
... Proposition 4.4. Given a covering space p : X̃ → X, a homotopy ft : Y → X and a lift f˜0 : Y → X̃ lifting of f0 , there exists a unique homotopy f˜t : Y → X̃ starting at f˜0 lifting ft . Proof. Let F : Y × I → X be our homotopy map. We first construct a lift F̃ : N × I → X̃ for N a neighborhood of a ...
... Proposition 4.4. Given a covering space p : X̃ → X, a homotopy ft : Y → X and a lift f˜0 : Y → X̃ lifting of f0 , there exists a unique homotopy f˜t : Y → X̃ starting at f˜0 lifting ft . Proof. Let F : Y × I → X be our homotopy map. We first construct a lift F̃ : N × I → X̃ for N a neighborhood of a ...
Introduction The notion of shape of compact metric
... witii the previously given notion**. * This paper has b:en written while the author was visiting the University of Heidelberg on leave from :he C nivehsity of Zagreb. It was presented at the Colloquium on Topology, Keszthely, Hung mm:‘, June 19-23, 1972. ** After thi.s papeA lrg1 been submitted, thr ...
... witii the previously given notion**. * This paper has b:en written while the author was visiting the University of Heidelberg on leave from :he C nivehsity of Zagreb. It was presented at the Colloquium on Topology, Keszthely, Hung mm:‘, June 19-23, 1972. ** After thi.s papeA lrg1 been submitted, thr ...
Combinatorial Equivalence Versus Topological Equivalence
... work of Stallings [13] related to this question, also. A weaker question may be asked, which, in the light of Theorem 6.1, is relevant. Are there two combinatorial imbeddings f,g:S,-2_Sm such that these bounded complements M, Mg c S'" have the same homotopy type but distinct simple homotopy type? Fi ...
... work of Stallings [13] related to this question, also. A weaker question may be asked, which, in the light of Theorem 6.1, is relevant. Are there two combinatorial imbeddings f,g:S,-2_Sm such that these bounded complements M, Mg c S'" have the same homotopy type but distinct simple homotopy type? Fi ...
Martin-Löf`s Type Theory
... The notion of constructive proof is closely related to the notion of computer program. To prove a proposition (∀x ∈ A)(∃y ∈ B)P (x, y) constructively means to give a function f which when applied to an element a in A gives an element b in B such that P (a, b) holds. So if the proposition (∀x ∈ A)(∃y ...
... The notion of constructive proof is closely related to the notion of computer program. To prove a proposition (∀x ∈ A)(∃y ∈ B)P (x, y) constructively means to give a function f which when applied to an element a in A gives an element b in B such that P (a, b) holds. So if the proposition (∀x ∈ A)(∃y ...
MA5209L4 - Maths, NUS - National University of Singapore
... Remark* E is pathwise connected, p is surjective. ...
... Remark* E is pathwise connected, p is surjective. ...
(pdf)
... groups are relatively easy to define, and are in some sense a more powerful tool than homology, but are in general not easily computed. The homology groups, which up to natural isomorphism can be constructed in a number of seemingly disparate ways, while perhaps a less powerful tool are much more ea ...
... groups are relatively easy to define, and are in some sense a more powerful tool than homology, but are in general not easily computed. The homology groups, which up to natural isomorphism can be constructed in a number of seemingly disparate ways, while perhaps a less powerful tool are much more ea ...
Homotopy type theory
In mathematical logic and computer science, homotopy type theory (HoTT) refers to various lines of development of intensional type theory, based on the interpretation of types as objects to which the intuition of (abstract) homotopy theory applies.This includes, among other lines of work, the construction of homotopical and higher-categorical models for such type theories; the use of type theory as a logic (or internal language) for abstract homotopy theory and higher category theory; the development of mathematics within a type-theoretic foundation (including both previously existing mathematics and new mathematics that homotopical types make possible); and the formalization of each of these in computer proof assistants.There is a large overlap between the work referred to as homotopy type theory, and as the univalent foundations project. Although neither is precisely delineated, and the terms are sometimes used interchangeably, the choice of usage also sometimes corresponds to differences in viewpoint and emphasis. As such, this article may not represent the views of all researchers in the fields equally.