proof-theory
Provides logical strategies and tools for verifying proofs in mathematical logic.
Install
mkdir -p .claude/skills/proof-theory && curl -L -o skill.zip "https://agentskills.codes/api/skills/download/5911" && unzip -o skill.zip -d .claude/skills/proof-theory && rm skill.zipInstalls to .claude/skills/proof-theory
Activation
This is the description your AI agent reads to decide when to run this skill — the better it matches your request, the more reliably it fires.
Problem-solving strategies for proof theory in mathematical logicKey capabilities
- →Apply structural induction to logical proofs
- →Execute Z3 theorem prover verification
- →Normalize proofs via cut elimination
- →Validate soundness and completeness of logic
- →Verify step-by-step mathematical reasoning
How it works
It utilizes the Z3 SMT solver and defined proof-theory scripts to programmatically evaluate the validity of logical propositions and induction proofs.
Inputs & outputs
When to use proof-theory
- →Perform structural induction on mathematical structures
- →Verify soundness and completeness of formal logical systems
- →Execute Z3-based proof verification
- →Normalize proof structures using cut elimination
About this skill
Proof Theory
When to Use
Use this skill when working on proof-theory problems in mathematical logic.
Decision Tree
-
Proof Strategy Selection
- Direct proof: assume premises, derive conclusion
- Proof by contradiction: assume negation, derive false
- Proof by cases: split on disjunction
- Induction: base case + inductive step
-
Structural Induction
- Define well-founded ordering on structures
- Base: prove for minimal elements
- Step: assume for smaller, prove for current
z3_solve.py prove "induction_principle"
-
Cut Elimination
- Gentzen's Hauptsatz: cuts can be eliminated
- Subformula property: only subformulas appear
- Useful for proof normalization
-
Completeness/Soundness Check
- Soundness: if provable then valid
- Completeness: if valid then provable
z3_solve.py prove "soundness_theorem"
-
Proof Verification
- Check each step follows from rules
- Verify dependencies are satisfied
math_scratchpad.py verify "proof_steps"
Tool Commands
Z3_Induction_Base
uv run python -m runtime.harness scripts/cc_math/z3_solve.py prove "P(0)"
Z3_Induction_Step
uv run python -m runtime.harness scripts/cc_math/z3_solve.py prove "ForAll([n], Implies(P(n), P(n+1)))"
Z3_Soundness
uv run python -m runtime.harness scripts/cc_math/z3_solve.py prove "Implies(derivable(phi), valid(phi))"
Math_Verify
uv run python -m runtime.harness scripts/cc_math/math_scratchpad.py verify "proof_structure"
Cognitive Tools Reference
See .claude/skills/math-mode/SKILL.md for full tool documentation.
When not to use it
- →Writing standard software business logic
- →Performance optimization of non-logical code
Prerequisites
Limitations
- →Only applicable to formal logical systems
- →Highly sensitive to correct formulation of premises
- →Z3 solving may time out on complex structural models
How it compares
It provides formal, computational verification of mathematical proofs instead of relying on manual derivation or fuzzy logic.
Compared to similar skills
proof-theory side by side with the closest alternatives in the catalog.
| Skill | Installs | Updated | Safety | Difficulty |
|---|---|---|---|---|
| proof-theory (this skill) | 1 | 7mo | Review | Advanced |
| tooluniverse-sdk | 0 | 2mo | Review | Advanced |
| literature-review | 559 | 2mo | Review | Advanced |
| openalex-database | 48 | 7mo | Review | Intermediate |
Try saying
Example prompts that trigger this skill in your AI assistant.
More by parcadei
View all by parcadei →You might also like
tooluniverse-sdk
mims-harvard
Build AI scientist systems using ToolUniverse Python SDK for scientific research. Use when users need to access 1000++ scientific tools through Python code, create scientific workflows, perform drug discovery, protein analysis, genomics analysis, literature research, or any computational biology task. Triggers include requests to use scientific tools programmatically, build research pipelines, analyze biological data, search literature, predict drug properties, or create AI-powered scientific workflows.
literature-review
K-Dense-AI
Conduct comprehensive, systematic literature reviews using multiple academic databases (PubMed, arXiv, bioRxiv, Semantic Scholar, etc.). This skill should be used when conducting systematic literature reviews, meta-analyses, research synthesis, or comprehensive literature searches across biomedical, scientific, and technical domains. Creates professionally formatted markdown documents and PDFs with verified citations in multiple citation styles (APA, Nature, Vancouver, etc.).
openalex-database
davila7
Query and analyze scholarly literature using the OpenAlex database. This skill should be used when searching for academic papers, analyzing research trends, finding works by authors or institutions, tracking citations, discovering open access publications, or conducting bibliometric analysis across 240M+ scholarly works. Use for literature searches, research output analysis, citation analysis, and academic database queries.
scientific-critical-thinking
davila7
Evaluate research rigor. Assess methodology, experimental design, statistical validity, biases, confounding, evidence quality (GRADE, Cochrane ROB), for critical analysis of scientific claims.
biorxiv-database
lifangda
Efficient database search tool for bioRxiv preprint server. Use this skill when searching for life sciences preprints by keywords, authors, date ranges, or categories, retrieving paper metadata, downloading PDFs, or conducting literature reviews.
physics-validator
omriwen
Validate optical physics parameters including Fresnel numbers, diffraction regimes, and resolution limits. This skill should be used when configuring Telescope, Microscope, or Camera instruments to ensure physically realistic parameters.