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KrystianYCSilva
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set-theory-implementation-patterns

Provides architectural designs for modeling complex physical and digital systems with UML and SysML diagrams.

Updated Feb 20, 2026
Search Tags:#set theory#formal methods#dual-mode#intensional sets#extensional sets#lazy evaluation#isomorphism testing
One-click install
npx skills add https://github.com/KrystianYCSilva/math-theory-lib --skill set-theory-implementation-patterns
Or copy as Structured Prompt for Agent▼
Please help me install this Agent Skill.
Skill: set-theory-implementation-patterns
Source: https://github.com/KrystianYCSilva/math-theory-lib/tree/main/.gemini/skills/set-theory-implementation-patterns
Command: npx skills add https://github.com/KrystianYCSilva/math-theory-lib --skill set-theory-implementation-patterns

SYSTEM DOCUMENTATION & REQUIREMENTS

What problem does it solve?

Representing abstract, potentially infinite mathematical sets and constructions on finite machines is challenging because implementations must balance formal correctness with performance and usability. This skill provides architectural patterns to model sets, operations, and number constructions so implementations remain practical while retaining a verifiable mathematical foundation.

Core Features & Use Cases

  • Extensional vs Intensional Representations: Model finite sets by explicit membership and infinite or comprehension-defined sets by predicates over a domain.
  • Dual-Mode Architecture: Separate a fast Kernel (practical, native-backed values) from a Construction mode (axiomatic, set-theoretic constructions) and use isomorphism testing to relate them.
  • Lazy Evaluation for Large/Infinite Sets: Represent power sets and cartesian products lazily to avoid materializing exponentially large or infinite collections.
  • Use Cases: Designing theorem provers, symbolic algebra libraries, verified numeric libraries, and educational tools that demonstrate formal constructions with performant runtime representations.

Quick Start

Show me Kotlin implementations for an ExtensionalSet of the first five natural numbers, an IntensionalSet for even numbers, and a LazyPowerSet sketch that generates subsets lazily.

Frequently Asked Questions about set-theory-implementation-patterns

High-intent search queries and answers about installing and using this skill.

FAQPage Schema
How do I implement set theory representations for infinite sets in software?▼

To implement infinite sets, use intensional representations that define membership via predicates over a domain, avoiding the need to materialize exponentially large collections in memory.

What is the best way to handle power set and cartesian product operations on large sets?▼

The best way to handle large power sets and cartesian products is lazy, on-demand evaluation, which generates subsets dynamically and prevents the materialization of exponentially large collections.

How does a dual-mode architecture help verify formal set theory constructions?▼

A dual-mode architecture separates a fast practical Kernel from an axiomatic Construction mode, using isomorphism testing to verify that runtime values match formal set-theoretic definitions.

When do I need to separate extensional and intensional sets in library design?▼

Separate extensional and intensional sets when your library must model both finite collections by explicit membership and comprehension-defined or infinite sets by predicates over a domain.

Can I use these set-theory implementation patterns for theorem provers and symbolic algebra systems?▼

Yes, these architectural patterns apply directly to designing theorem provers, symbolic algebra systems, and verified numeric libraries that require performant runtime representations with mathematical correctness.

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