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bijections_and_isomorphisms

Construct bijections or isomorphisms between algebraic and combinatorial structures in Lean 4.

1|Updated Mar 14, 2026
Search Tags:#lean4#theorem-proving#proof#bijections#isomorphisms#mathlib4
One-click install
npx skills add https://github.com/bneb/perqed --skill bijections-and-isomorphisms
Or copy as Structured Prompt for Agent▼
Please help me install this Agent Skill.
Skill: bijections_and_isomorphisms
Source: https://github.com/bneb/perqed/tree/main/.agents/skills/bijections_and_isomorphisms
Command: npx skills add https://github.com/bneb/perqed --skill bijections-and-isomorphisms

SYSTEM DOCUMENTATION & REQUIREMENTS

What problem does it solve?

Prove that two mathematical structures are essentially identical by exhibiting a structure-preserving map with an inverse; used to transfer properties and count equivalences.

Core Features & Use Cases

  • Establish a bijection or isomorphism between mathematical structures (sets, groups, graphs, rings) to transfer properties.
  • Provide explicit forward and inverse maps and prove injectivity and surjectivity to certify equivalence.
  • Apply within Lean 4 / mathlib4 formalizations to reuse results and streamline proofs across related objects.

Quick Start

Define a forward map f: A → B and prove it is bijective to establish the isomorphism.

Frequently Asked Questions about bijections_and_isomorphisms

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

FAQPage Schema
How do I prove a bijection between two algebraic structures in Lean 4?▼

To prove a bijection in Lean 4, define an explicit forward map and its inverse, then prove injectivity and surjectivity to establish structural equivalence. This certifies that properties can be transferred between the two mathematical objects.

What is the best way to transfer properties across isomorphic structures in mathlib4?▼

Transferring properties across isomorphic structures in mathlib4 requires constructing a structure-preserving map with a verified inverse. By proving this equivalence, you can safely reuse established theorems and results across related algebraic or combinatorial objects.

Can I use this approach to prove equivalence for combinatorial graphs?▼

Yes, this approach applies to combinatorial graphs and other algebraic structures like sets, groups, and rings. Providing explicit forward and inverse maps with injectivity and surjectivity proofs establishes their structural equivalence.

When do I need to construct an isomorphism rather than a simple bijection?▼

You need an isomorphism rather than a simple bijection when transferring properties requires preserving the underlying algebraic or combinatorial structure. This ensures the structural operations remain consistent across the mapped objects in your Lean 4 formalization.

Does establishing an isomorphism in Lean 4 require explicit inverse maps?▼

Yes, establishing an isomorphism in Lean 4 requires explicit forward and inverse maps. You must also provide proofs of injectivity and surjectivity to certify the mathematical equivalence and ensure compatibility with mathlib4.

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