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Implementing Mathematics with The Nuprl Proof Development System

Coq is a proof assistant based on a higher-order logic allowing powerful definitions of functions. Coq V6.1 is available by anonymous ftp at ftp.inria.fr:/INRIA/Projects/coq/V6.1 and ftp.ens-lyon.fr:/pub/LIP/COQ/V6.1

https://hal.inria.fr/inria-00069968/document

The Coq proof assistant : reference manual, version 6.1

This book represents the first treatment of computable analysis at the graduate level within the tradition of classical mathematical reasoning. Among the topics dealt with are: classical analysis, Hilbert and Banach spaces, bounded and unbounded linear operators, eigenvalues, eigenvectors, and equations of mathematical physics. The book is self-contained, and yet sufficiently detailed to provide an introduction to research in this area.

Computability in analysis and physics

Part 1 Polymorphic sets: the semantics of the judgement forms general rules enumeration sets Cartesian product of a family of sets equality sets natural numbers lists cartesian product of two sets disjoint union of two sets disjoint union of a family of sets the set of small sets (the first universe) well-orderings general trees. Part 2 Subsets: subsets in the basic set theory the subset theory. Part 3 Monomorphic sets: types defining sets in terms of types. Part 4 Examples: some small examples program derivation specification of abstract data types.

Programming in Martin-Lo¨f's type theory: an introduction

data type, 179 abstraction, 14 Abstraction rule, 143 absurdity, 43 append, 68 application, 13 Application rule, 143 apply, 48, 148 arity of an expression, 18 Assumption rule, 123, 142 AUTOMATH, 8 axiom of choice, 115 Bishop, 6 Bool, 44 Bool rules, 44–45 Boolean, 44–45 Calculus of Constructions, 8 canonical expression, 26 cartesian product of a family of sets, 47–55, 148 case{i1,,in}, 41 case, 151 Church, 17 combination, 15 combined expression, 17 conjunction, 74 cons, 67, 152 Constable, 6, 167 constructive mathematics, 6 context, 29, 139 Coquand, 3, 7, 8 Curry, 10 Curry-Howard interpretation, 6 Cut rule for elements in sets, 124 Cut rule for equal elements in sets, 124 Cut rule for equal sets, 123 Cut rule for propositions, 123 Cut rule for true propositions, 124 de Bruijn, 8, 10 definiendum, 19 definiens, 19 definitional equality of expressions, 15 Dijkstra, 167, 171 disjoint union of a family of sets, 81– 82, 149 disjoint union of two sets, 79–80, 150 Dummet, 6 El formation rule, 139 elimination rule, 35 empty set, 43 enumeration sets, 41–42, 151 Eq, 57 Eq rules, 60–61 equality as a proposition, 57, 117 between canonical elements, 27 between elements in a set, 29, 31, 37, 119, 121 between expressions, 15 between sets, 28, 30, 37, 118, 121 extensional, 57 identity between objects in a type, 138 identity between types, 138 intensional, 57 equality rule, 35 equality sets, 57–62, 150 examples: Bool has only the elements true and false, 161 A partitioning problem, 171 Decidable predicates, 162 Division by 2, 155 Even or odd, 159 Module for sets with a computable equality, 182 Stack of A elements, 182 Stack of natural numbers, 179 Stronger elimination rules, 163 existential quantifier, 81, 82 expressions, 13 arity, 18 canonical —, 26 combined —, 17 saturated —, 17 single —, 17 unsaturated —, 17 extensional equality, 57 wfalse, 44 Fermat’s last theorem, 4 formation rule, 35 Frege, 17

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Programming in Martin-Löf's Type Theory

Foundations of Constructive Mathematics

One. Practice and Philosophy of Constructive Mathematics.- I. Examples of Constructive Mathematics.- 1. The Real Numbers.- 2. Constructive Reasoning.- 3. Order in the Reals.- 4. Subfields of R with Decidable Order.- 5. Functions from Reals to Reals.- 6. Theorem of the Maximum.- 7. Intermediate Value Theorem.- 8. Sets and Metric Spaces.- 9. Compactness.- 10. Ordinary Differential Equations.- 11. Potential Theory.- 12. The Wave Equation.- 13. Measure Theory.- 14. Calculus of Variations.- 15. Plateau's Problem.- 16. Rings, Groups, and Fields.- 17. Linear Algebra.- 18. Approximation Theory.- 19. Algebraic Topology.- 20. Standard Representations of Metric Spaces.- 21. Some Assorted Problems.- II. Informal Foundations of Constructive Mathematics.- 1. Numbers.- 2. Operations or Rules.- 3. Sets and Presets.- 4. Constructive Proofs.- 5. Witnesses and Evidence.- 6. Logic.- 7. Functions.- 8. Axioms of Choice.- 9. Ways of Constructing Sets.- 10. Definite Presets.- III. Some Different Philosophies of Constructive Mathematics.- 1. The Russian Constructivists.- 2. Recursive Analysis.- 3. Bishop's Constructivism.- 4. Objective Intuitionism.- 5. Sets in Intuitionism.- 6. Brouwerian Intuitionism.- 7. Martin-Lof s Philosophy.- 8. Church's Thesis.- IV. Recursive Mathematics: Living with Church's Thesis.- 1. Constructive Recursion Theory.- 2. Diagonalization and "Weak Counterexamples".- 3. Continuity of Effective Operations.- 4. Specker Sequences.- 5. Failure of Konig's Lemma: Kleene's Singular Tree.- 6. Singular Coverings.- 7. Non-Uniformly Continuous Functions.- 8. The Infimum of a Positive Function.- 9. Theorem of the Maximum Revisited.- 10. The Topology of the Disk in Recursive Mathematics.- 11. Pointwise Convergence Versus Uniform Convergence.- 12. Connectivity of Intervals.- 13. Another Surprise in Recursive Topology.- 14. A Counterexample in Descriptive Set Theory.- 15. Differential Equations with no Computable Solutions.- V. The Role of Formal Systems in Foundational Studies.- 1. The Axiomatic Method.- 2. Informal Versus Formal Axiomatics.- 3. Adequacy and Fidelity: Criteria for Formalization.- 4. Constructive and Classical Mathematics Compared.- 5. Arithmetic of Finite Types.- 6. Formalizing Constructive Mathematics in HA?.- Two. Formal Systems of the Seventies.- VI. Theories of Rules.- 1. The Logic of Partial Terms.- 2. Combinatory Algebras.- 3. Axiomatizing Recursive Mathematics.- 4. Term Reduction and the Church-Rosser Theorem Ill.- 5. Combinatory Logic and ?-Calculus.- 6. Term Models.- 7. Continuous Models.- 8. Finite Type Structures and Continuity in Combinatory Algebras.- 9. Set-Theoretic and Topological Models.- 10. Discussion: Adequacy and Fidelity of EON?.- VII. Readability.- 1. Definition and Soundness of Realizability.- 2. Realizability and Models.- 3. Some Simple Applications.- 4. Existence Properties.- 5. q-Realizability.- 6. Rules of Choice.- 7. Discussion: Numerical Meaning.- VIII. Constructive Set Theories.- 1. Intuitionistic Zermelo-Fraenkel Set Theory, IZF.- 2. Non-Extensional Set Theory.- 3. The Double-Negation Interpretation for IZF.- 4. Realizability for Set Theory Without Extensionality.- 5. Realizability and Models.- 6. Realizability for IZF.- 7. Connection with Realizability for Arithmetic.- 8. Consistency of Church's Thesis with IZF.- 9. The Numerical Existence Property for IZF.- 10. Discussion.- 11. The Theory B.- 12. More Discussion.- 13. Intermediate Constructive Set Theories.- IX. The Existence Property in Constructive Set Theory.- 1. Introduction.- 2. The Set Existence Property for HAS.- 3. The Existence Property in Set Theories.- X. Theories of Rules, Sets, and Classes.- 1. The Theory FML.- 2. The Theories EM0 and EM0 +J.- 3. Models of Feferman's Theories.- 4. Realizability.- 5. The Axiom of Choice.- 6. q-Realizability.- 7. Term Existence Property.- 8. Evaluation of Numerical Terms.- 9. Numerical Existence Property.- 10. Decidable Equality.- 11. Extensionality in Feferman's Theories.- 12. Some Remarks on Formalizing Mathematics in FML + DC.- 13. Functions, Operations, and Axioms of Choice.- 14. The Theory SOF (Sets, Operations, Functions).- 15. Some Theorems of SOF.- 16. Realizability for SOF.- 17. Discussion.- XI. Constructive Type Theories.- 1. Forms of Judgment.- 2. Philosophical Remarks on Sets, Categories, and Canonical Elements.- 3. Hypothetical Judgments.- 4. Families of Sets.- 5. Disjoint Union and Existential Quantification.- 6. Abstraction.- 7. Cartesian Product and Conjunction.- 8. The Constant E and Projection Functions.- 9. Product and Universal Quantification.- 10. Implication.- 11. N, Nk, and R.- 12. Disjoint Union.- 13. The I-Rules.- 14. How Martin-Lof's Rules Actually Define a Formal System.- 15. List of Rule Schemata of Martin-Lof's System ML0.- 16. Other Formulations of ML0.- 17. Interpretation of HA? + AC + EXT in ML0.- 18. Formalizing Mathematics in ML0.- 19. Is the Theory HA? + EXT + AC Constructive?.- 20. The Realizability Model of ML0.- 21. Martin-Lof's Universes.- 22. The Arithmetic Theorems of Martin-Lof's Systems, and a New Realizability for Arithmetic.- 23. Discussion.- Three. Metamathematical Studies.- XII. Constructive Models of Set Theory.- 1. Aczel's "Iterative Sets".- 2. Interpreting Subtheories of Intuitionistic ZF in Feferman's Theories.- XIII. Proof-Theoretic Strength.- 1. Definitions About Proof-Theoretic Strength.- 2. ?1/1-AC and ID1-.- 3. The Strength of Feferman's Theory EM0 + J and Related Theories.- 4. The Strength of Constructive Set Theory T2.- 5. The Strength of Martin-Lof's Theories.- 6. Theories with the Strength of Arithmetic.- 7. Conservative Extension Theorems.- XIV. Some Formalized Metamathematics and Church's Rule.- 1. The Theories Tb and Ta.- 2. Formalization of Normal-Term Arguments.- 3. Formalized Models.- 4. Church's Rule.- 5. Truth Definitions and Reflection Principles.- 6. Formalized Realizability and Existence Properties.- 7. Discussion.- XV. Forcing.- 1. Ordinary Forcing.- 2. Conservative Extension Results.- 3. Uniform Forcing.- XVI. Continuity.- 1. Continuity Principles.- 2. Continuity and Church's Thesis.- 3. Consistency of Brouwer's Principle and Uniform Continuity.- 4. Derived Rules Related to Continuity.- Four. Metaphilosophical Studies.- XVII. Theories of Rules and Proofs.- 1. Review of the Relevant Literature.- 2. The Main Issues.- 3. A Theory of Rules and Proofs.- 4. Undecidability of the Proof-Predicate in C.- 5. Consistency of C.- 6. Discussion.- 7. Frege Structures.- 8. Existence of Frege Structures.- 9. Set Theory and Frege Structures.- 10. A Theory of Rules, Proofs, and Sets.- Historical Appendix.- 1. From Gauss to Zermelo: The Origins of Non-Constructive Mathematics.- 2. From Kant to Hilbert: Logic and Philosophy.- 3. Brouwer and the Dutch Intuitionists.- 4. Early Formal Systems for Intuitionism.- 5. Kleene: The Marriage of Recursion Theory and Intuitionism.- 6. The Russian Constructivists and Recursive Analysis.- 7. Model Theory of Intuitionistic Systems.- 8. Logical Studies of Intuitionistic Systems.- 9. Bishop and his Followers.- 10. The Latest Decade.- References.- Index of Axioms, Abbreviations, and Theories.- Index of Names.- Index of Symbols.

Foundations of Constructive Mathematics: Metamathematical Studies

This paper lists eight design criteria that must be met if we are to provide successful computer support for education in algebra, trigonometry, and calculus. It also describes Mathpert, a piece of software that was built with these criteria in mind. The description given here is intended for designers of other software, for designers of new teaching materials and curricula utilizing mathematical software, and for professors interested in using such software. The design principles in question involve both the user interface and the internal operation of the software. For example, three important principles are cognitive fidelity, the glass box principle, and the correctness principle. After an overview of design principles, we discuss the design of Mathpert in the light of these principles, showing how the main lines of the design were determined by these principles. (The scope of this paper is strictly limited to an exposition of the design principles and their application to Mathpert. I shall not attempt to review projects other than Mathpert in the light of these design principles.)

Design principles of Mathpert: software to support education in algebra and calculus

Publisher Summary Progress in applications to large-scale computer systems, depends on the design of new formal systems. This chapter raises some issues and makes a technical contribution by considering some theories of Feferman from the viewpoint of computer science, and comparing them with the theories of Martin-Lőf. The command language and assertion language of proving programs are discussed in the chapter. By “proving programs” is meant more explicitly: proving properties of programs or proving properties of the execution of programs. People speak of the “correctness” of a program with respect to its “specifications”—by this they mean that if the program gets an input of the kind it is designed for, it will produce an output having certain specified relations with the input. This notion is divided into the two notions of “total correctness” (an output is always produced and it is correct) and “partial correctness” (if an output is produced then it is correct).

Michael Beeson

Papers

Foundations of Constructive Mathematics

Foundations of Constructive Mathematics: Metamathematical Studies

Design principles of Mathpert: software to support education in algebra and calculus

Proving Programs and Programming Proofs

Recursive models for constructive set theories