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Realizability interpretation of proofs in constructive analysis
, 2006
"... We prove constructively (in the style of Bishop) that every monotone continuous function with a uniform modulus of increase has a continuous inverse. The proof is formalized, and a realizing term extracted. It turns out that even in the logical term language – a version of Gödel’s T – evaluation is ..."
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We prove constructively (in the style of Bishop) that every monotone continuous function with a uniform modulus of increase has a continuous inverse. The proof is formalized, and a realizing term extracted. It turns out that even in the logical term language – a version of Gödel’s T – evaluation is reasonably efficient.
The constructive reals as a Java Library
 J. Log. Algebr. Program
, 2004
"... We describe an implementation of the computable (or constructive) real numbers as a pure Java library. To the user, the library interface appears very similar to that of some other numeric types provided by the standard Java library. The primary goal of the implementation is simplicity, so that the ..."
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We describe an implementation of the computable (or constructive) real numbers as a pure Java library. To the user, the library interface appears very similar to that of some other numeric types provided by the standard Java library. The primary goal of the implementation is simplicity, so that the implementation could be easily understood, and to allow simple informal correctness arguments. We hope to demonstrate that even such a basic implementation of constructive real arithmetic can be useful in a number of contexts, including in a desk calculator utility distributed with the package. A secondary goal was to demonstrate that some secondorder functions on the reals, such as restricted inverse and derivative operations, can be implemented with su#cient performance to be useful.
Constructive Analysis with Witnesses
"... Contents 1. Real Numbers 3 2 3 1.2. Reals, Equality of Reals 5 1.3. The Archimedian Axiom 6 1.4. Nonnegative and Positive Reals 6 1.5. Arithmetical Functions 7 1.6. Comparison of Reals 8 1.7. NonCountability 10 1.8. Cleaning of Reals 11 2. Sequences and Series of Real Numbers 11 2.1. Completenes ..."
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Contents 1. Real Numbers 3 2 3 1.2. Reals, Equality of Reals 5 1.3. The Archimedian Axiom 6 1.4. Nonnegative and Positive Reals 6 1.5. Arithmetical Functions 7 1.6. Comparison of Reals 8 1.7. NonCountability 10 1.8. Cleaning of Reals 11 2. Sequences and Series of Real Numbers 11 2.1. Completeness 11 2.2. Limits and Inequalities 13 2.3. Series 13 2.4. Redundant Dyadic Representation of Reals 14 2.5. Convergence Tests 15 2.6. Reordering Theorem 17 2.7. The Exponential Series 18 3. The Exponential Function for Complex Numbers 21 4. Continuous Functions 23 4.1. Suprema and In ma 24 4.2. Continuous Functions 25 4.3. Application of a Continuous Function to a Real 27 4.4. Continuous Functions and Limits 28 4.5. Composition of Continuous Functions 28 4.6. Properties of Continuous Functions 29 4.7. Intermediate Value Theorem 30 4.8. Continuity of Functions with More Than One Variable 32 5. Dierentiation 33 5.1. Derivatives 33 5.2. Bounds on the Slope 33 5.3. Properties of Derivatives 34 5
Program extraction in constructive analysis. Submitted to: Logicism, Intuitionism, and Formalism – What has become of them
, 2006
"... We sketch a development of constructive analysis in Bishop’s style, with special emphasis on low typelevel witnesses (using separability of the reals). The goal is to set up things in such a way that realistically executable programs can be extracted from proofs. This is carried out for (1) the Int ..."
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We sketch a development of constructive analysis in Bishop’s style, with special emphasis on low typelevel witnesses (using separability of the reals). The goal is to set up things in such a way that realistically executable programs can be extracted from proofs. This is carried out for (1) the Intermediate Value Theorem and (2) the existence of a continuous inverse to a monotonically increasing continuous function. Using the Minlog proof assistant, the proofs leading to the Intermediate Value Theorem are formalized and realizing terms extracted. It turns out that evaluating these terms is a reasonably fast algorithm to compute, say, approximations of √ 2. 1