We present a dependent Hoare Type Theory (HTT) which provides support for reasoning about programs with higherorder functions and effects, including non-termination, state with aliasing and pointer arithmetic. The type structure encapsulates effectful commands using a monad indexed by preand post-conditions in the style of Hoare logic. The theory carefully distinguishes between an appropriate notion of definitional equality and propositional equality, in order to maintain the relative decidability of

methods

Proof Theory

Dependent type theory of stateful higher-order functions

The paper outlines an interpretation of one of the most important and original contributions of David Hilbert’s monograph Foundations of Geometry (1899), namely his internal arithmetization of geometry. It is claimed that Hilbert’s profound interest in the problem of the introduction of numbers into geometry responded to certain epistemological aims and methodological concerns that were fundamental to his early axiomatic investigations into the foundations of elementary geometry. In particular, it is shown that a central concern that motivated Hilbert’s axiomatic investigations from very early on was the aim of providing an independent basis for geometry. Accordingly, these concerns about an independent grounding for elementary geometry determined very clear methodological constraints in the process of embedding it into a formal axiomatic system. It is argued that Hilbert not only sought to show that geometry could be considered a pure mathematical theory, once it was presented as a formal axiomatic system; he also aimed at showing that in the construction of such an axiomatic system one could proceed purely geometrically, avoiding concept formations borrowed from other mathematical disciplines like arithmetic or analysis.

Bridging the gap between analytic and synthetic geometry: Hilbert’s axiomatic approach

Apologies. The purpose of the following talk is to give an overview of the present state of aims, methods and results in Pure Proof Theory. Shortage of time forces me to concentrate on my very personal views. This entails that I will emphasize the work which I know best, i.e., work that has been done in the triangle Stanford, Munich and Munster. I am of course well aware that there are as important results coming from outside this triangle and I apologize for not displaying these results as well. Moreover the audience should be aware that in some points I have to oversimplify matters. Those who complain about that are invited to consult the original papers. 1.1. General. Proof theory startedwithHilbert's Programme which aimed at a finitistic consistency proof for mathematics. By Godel's Theorems, however, we know that we can neither formalize all mathematics nor even prove the consistency of formalized fragments by finitistic means. Inspite of this fact I want to give some reasons why I consider proof theory in the style of Gentzen's work still as an important and exciting field of Mathematical Logic. I will not go into applications of Gentzen's cut-elimination technique to computer science problems—this may be considered as applied proof theory—but want to concentrate on metamathematical problems and results. In this sense I am talking about Pure Proof Theory . Mathematicians are interested in structures. There is only one way to find the theorems of a structure. Start with an axiom system for the structure and deduce the theorems logically. These axiom systems are the objects of proof-theoretical research. Studying axiom systems there is a series of more or less obvious questions.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S1079898600008738

Pure Proof Theory Aims, Methods and Results

Gentzen's sequential system LJ of intuitionistic logic has two symbols of implication. One is the logical symbol → and the other is the metalogical symbol ⇒ in sequentsConsidering the logical system LJ as a mathematical object, we understand that the logical symbols ∧, ∨, →, ¬, ∀, ∃ are operators on formulas, and ⇒ is a relation. That is, φ ⇒ Ψ is a metalogical sentence which is true or false, on the understanding that our metalogic is a classical logic. In other words, we discuss the logical system LJ in the classical set theory ZFC, in which φ ⇒ Ψ is a sentence.The aim of this paper is to formulate an intuitionistic set theory together with its metatheory. In Takeuti and Titani [6], we formulated an intuitionistic set theory together with its metatheory based on intuitionistic logic. In this paper we postulate that the metatheory is based on classical logic.Let Ω be a cHa. Ω can be a truth value set of a model of LJ. Then the logical symbols ∧, ∨, →, ¬, ∀x, ∃x are interpreted as operators on Ω, and the sentence φ ⇒ Ψ is interpreted as 1 (true) or 0 (false). This means that the metalogical symbol ⇒ also can be expressed as a logical operators such that φ ⇒ Ψ is interpreted as 1 or 0.

Completeness of global intuitionistic set theory

The meaning of a formula built out of proof-functional connectives depends in an essential way upon the intensional aspect of the proofs of the component subformulas. We study three such connectives, strong equivalence (where the two directions of the equivalence are established by mutually inverse maps), strong conjunction (where the two components of the conjunction are established by the same proof) and relevant implication (where the implication is established by an identity map). For each of these connectives we give a type assignment system, a realizability semantics, and a completeness theorem. This form of completeness implies the semantic completeness of the type assignment system.

Proof Theory

Citations

Dependent type theory of stateful higher-order functions

Bridging the gap between analytic and synthetic geometry: Hilbert’s axiomatic approach

Pure Proof Theory Aims, Methods and Results

Completeness of global intuitionistic set theory

Proof-functional connectives and realizability

Related Papers (5)

Untersuchungen über das logische Schließen. II

Introduction to Metamathematics

The collected papers of Gerhard Gentzen

Basic proof theory

Linear logic