Certainty by Construction
Certainty by Construction
Software and Mathematics in Agda
About the Book
Agda is not just a programming language; it's a completely new way of thinking about software, and mathematical objects at large. This book is on successfully wielding Agda: teaching everything you need to know to get productive in modeling problems and in verifying your solutions.
About the Author
Table of Contents
- Preface
-
The Co-blub Paradox
- A World Without Execution?
-
Introduction to Agda
- Modules and Imports
- Syntax Highlighting
- Types
- Dealing with Unicode
- A Note on Syntax
- Operators
- Agda’s Computational Model
- Records and Tuples
- Function Types
- The Curry/Uncurry Isomorphism
- Implicit Arguments
- Wrapping Up
-
An Exploration of Numbers
- Natural Numbers
- A Note on Algebraic Data Types
- Playing with Naturals
- Induction
- Two Notions of Evenness
- Constructing Evidence
- Addition
- Termination Checking
- Multiplication and Exponentiation
- Semi-subtraction
- Integers
-
Proof Objects
- Constructivism
- Propositions as Types
- Hard to Prove or Simply False?
- The Equality Type
- Congruence
- Identities
- Symmetry and Involutivity
- Transitivity
- Mixfix Parsing
- Equational Reasoning
- Ergonomics, Associativity and Commutativity
- Facts About Natural Numbers
-
Decidability
- Negation
- Bottom
- Inequality
- Negation Considered as a Callback
- Intransitivity of Inequality
- Decidability
- Binary Trees
- Binary Search Trees
- Insertion into BSTs
- Intrinsic vs Extrinsic Proofs
- An Intrinsic BST
-
Quotients
- Setoids
- Constructions on Setoids
-
Functions, Big and Small
- Matrices
- The Row-Major Representation
- Function Representation
- Matrices as Functions
-
Ring Solving
- Canonical Forms
- Sketching Out a Ring Solver
-
Isomorphism
- More Setoids
- Formalizing Isomorphisms
- Injectivity
- Surjectivity
- Showing the Bijection
- Moving Forwards
-
Algebraic Structures
- More Monoids
- Monoid Homomorphisms
- Semigroups
-
Countability
- Finite Types
-
Partially Ordered Sets
- Examples of Posets
- The Poset of Substrings
- Meets and Joins
- Duality
-
Theorems about Monoids
- Free Constructions
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