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theorem-proving-and-counterexamples

Formalize and verify mathematical problems using ITPs and SMT solvers.

1|Updated Feb 24, 2026
Search Tags:#z3#lean4#theorem-proving#formalization#coq#isabelle#proof-assistance
One-click install
npx skills add https://github.com/dzackgarza/ai --skill theorem-proving-and-counterexamples
Or copy as Structured Prompt for Agent▼
Please help me install this Agent Skill.
Skill: theorem-proving-and-counterexamples
Source: https://github.com/dzackgarza/ai/tree/main/opencode/skills/theorem-proving-and-counterexamples
Command: npx skills add https://github.com/dzackgarza/ai --skill theorem-proving-and-counterexamples

SYSTEM DOCUMENTATION & REQUIREMENTS

💡 This Skill includes references (resource) components.

What problem does it solve?

Formalize complex mathematical problems and verify proofs across interactive theorem provers, automated theorem provers, SMT solvers, and specialized CAS, enabling rigorous reasoning beyond manual proofs.

Core Features & Use Cases

  • Lean 4 / Mathlib: formalization of contemporary mathematics across algebra, analysis, topology, and number theory.
  • Coq (Rocq): foundational proofs and software verification.
  • Isabelle/HOL: verification with Sledgehammer automation and AFP library support.
  • HOL Light: compact kernel for critical formalizations.
  • Prover9 / Mace4: first-order logic, finite models, and equational reasoning.
  • Z3 / SMT: arithmetic, optimization, and quantified theories.
  • PySAT / MiniZinc: combinatorial solving and constraint modeling.
  • GAP / PARI / M2: specialized CAS for algebra and number theory.

Quick Start

Describe a concrete math problem you want formalized and choose a toolchain (e.g., Lean 4 or Prover9) to begin.

Frequently Asked Questions about theorem-proving-and-counterexamples

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

FAQPage Schema
How do I formalize a mathematical proof in Lean 4 or Coq?▼

You can search for counterexamples using model-finding and computational algebra tools like Prover9, Mace4, Z3, and PySAT. This Skill applies automated theorem provers and SMT solvers to systematically search for counterexamples in mathematical problems.

Does this theorem proving workflow support Z3 and SMT solvers?▼

For first-order logic and equational reasoning, Prover9 and Mace4 are the preferred tools. This Skill supports these automated theorem provers for finding finite models and verifying first-order logic statements as part of its model-finding workflows.

Can I use GAP or PARI for number theory formalization?▼

This Skill handles both interactive theorem proving with Lean 4 and Coq, and automated theorem proving with Prover9 and Z3. It combines ITPs for foundational formalization with ATPs and SMT solvers for automated verification and counterexample discovery.

What's the best way to verify a topology proof using Isabelle/HOL?▼

The best way to verify a topology proof is using Isabelle/HOL with Sledgehammer automation and AFP library support. This Skill enables formal verification of mathematical problems in topology through its comprehensive interactive theorem proving workflows.

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