skills.rest
bneb
bnebCOMMUNITY
Explore all skills by this author →
GitHub

fixed_point_arguments

Determine fixed-point existence using Banach, Brouwer, or Tarski theorems.

1|Updated Mar 14, 2026
Search Tags:#lean#math#theorem#fixed-point#banach#brouwer#tarski
One-click install
npx skills add https://github.com/bneb/perqed --skill fixed-point-arguments
Or copy as Structured Prompt for Agent▼
Please help me install this Agent Skill.
Skill: fixed_point_arguments
Source: https://github.com/bneb/perqed/tree/main/.agents/skills/fixed_point_arguments
Command: npx skills add https://github.com/bneb/perqed --skill fixed-point-arguments

SYSTEM DOCUMENTATION & REQUIREMENTS

What problem does it solve?

Prove existence of x* such that f(x*) = x* for functions f : X → X using Brouwer's, Banach's, or Tarski's fixed-point theorems, based on the nature of the underlying space.

Core Features & Use Cases

  • Theorem selection: Choose Banach, Brouwer, or Tarski according to the space (complete metric space, compact convex set, or complete lattice) to guarantee existence of fixed points.
  • Constructive/Non-constructive outcomes: Obtain constructive iterations via Banach (when contraction) or non-constructive existence via Brouwer; leverage monotone operators via Tarski.
  • Use Case: Prove that a given iterative map has a fixed point, enabling formal verification, analysis proofs, or Lean-based developments of fixed-point arguments.

Quick Start

State the function f and the applicable space X, then apply the appropriate fixed-point theorem to establish the existence of x* with f(x*) = x*.

Frequently Asked Questions about fixed_point_arguments

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

FAQPage Schema
How do I prove a fixed point exists for a function in Lean?▼

To prove a fixed point exists, you state the function f and the space X, then apply Banach's, Brouwer's, or Tarski's fixed-point theorem based on your space's properties like contraction, continuity, or monotonicity.

When should I use Banach versus Brouwer or Tarski for a fixed-point proof?▼

Use Banach for complete metric spaces with contraction mappings, Brouwer for compact convex sets with continuous functions, and Tarski for complete lattices with monotone operators to establish fixed-point existence.

Can I get a constructive fixed point iteration using Banach's theorem in formal verification?▼

Yes, applying Banach's fixed-point theorem to a contraction mapping on a complete metric space yields a constructive iteration, providing an explicit sequence converging to the fixed point for formal verification.

What properties of a function do I need to specify to apply a fixed-point theorem?▼

You must explicitly specify whether the function is a contraction, continuous, or monotone, along with the underlying space type, so the appropriate fixed-point theorem can be selected and applied.

Does this approach support non-constructive existence proofs for continuous maps on compact convex sets?▼

Yes, applying Brouwer's fixed-point theorem to a continuous function on a compact convex set provides a non-constructive proof guaranteeing the existence of a fixed point without yielding an explicit iteration.

Related Skills

Explore More Education & Research Skills→
plurigridplurigrid
60

yoneda-directed

Apply directed Yoneda induction to prove properties along morphisms in Segal types.

Official
Advanced
0bserver070bserver07
3

formalize-informal-proof

Convert informal mathematical proofs into Lean 4 code for formal verification.

Community
Advanced
my04337my04337

lean-proof-planning

Plan and pre-validate Lean 4 proof strategies before theorem proving.

Community
Intermediate
bnebbneb
1

epsilon_delta_bounding

Construct explicit δ(ε) or N witnesses for Lean 4 / mathlib4 limit proofs.

Community
Advanced