proof-techniques
Proof writing and verification techniques for mathematical reasoning. Covers 13 proof strategies with canonical worked examples — direct, contrapositive, contradiction, cases, WLOG, vacuous/trivial, biconditional, counterexample, existence (constructive and non-constructive), uniqueness, induction (weak, strong, structural), pigeonhole/double-counting, and diagonal argument/invariants. Use when writing, verifying, or selecting proof strategies for any mathematical claim.
npx skills add majiayu000/claude-skill-registry --skill proof-techniques --agent claude-code
Same command for any agent — swap --agent for codex, cursor, copilot.
Weekly change comes from our own snapshots, not the repository page — it measures attention, not adoption.
What it does
This skill lists and explains 13 proof strategies used in mathematical reasoning. It provides concrete patterns, logical basis, worked examples, and guidance on when to apply each technique (direct proof, contrapositive, contradiction, cases, WLOG, vacuous/trivial proofs, biconditional proofs, counterexamples, existence proofs (constructive and non-constructive), uniqueness proofs, induction (weak, strong, structural), pigeonhole/double counting, and diagonal arguments/invariants). It aims to assist in writing, verifying, or selecting proof strategies for mathematical claims.
How it works
For each technique, the skill presents:
- Pattern: the general form to prove a statement of the type, and the typical reasoning flow.
- Logical basis: the foundational logical principle behind the technique.
- Worked example: a concrete instance demonstrating the technique.
- When to use: guidance on appropriate contexts or scenarios where the technique is effective.
- When it stalls / caveats: notes on limitations or considerations when a method may not be suitable.
The sections cover the 13 techniques in order, including Direct Proof, Proof by Contrapositive, Proof by Contradiction, Proof by Cases, Without Loss of Generality, Vacuous and Trivial Proofs, Biconditional Proof, Counterexample, Existence Proofs (Constructive and Non-Constructive), Uniqueness Proofs, Induction (Weak, Strong, Structural), Pigeonhole Principle and Double Counting, and Diagonal Arguments and Invariants.
When to use it
Use when deciding which proof strategy to apply to a mathematical claim, or to verify that a proposed proof follows the appropriate technique, or to learn/read canonical examples that illustrate each method.
What it can touch
The skill references specific proof techniques, examples, and logical steps. It relies on textual descriptions and mathematical patterns. No external tools are invoked within the included material beyond the general concept mappings; it does not specify executable commands or file-level interactions.
Caveats
Contains canonical examples and guidance; outcomes are not guaranteed by the skill and depend on correct application of each technique. Do not assume applicability beyond the patterns and conditions stated in each technique's description.
# Proof Techniques Mathematical proof is the mechanism by which conjectures become theorems. A proof is a finite sequence of logical steps, each justified by axioms, definitions, or previously established results, that establishes the truth of a mathematical statement beyond doubt. This skill catalogs 13 proof techniques with canonical worked examples, strategy selection heuristics, and readability guidance. **Agent affinity:** euclid (geometric and structural proofs), polya (strategy selection) **Concept IDs:** math-equations-expressions, math-systems-polynomials, math-pattern-recognition ## The Proof Toolbox at a Glance | # | Technique | Best for | Key signal | |---|---|---|---| | 1 | Direct proof | Default for implications | Can build Q from P step by step | | 2 | Contrapositive | P is hard to use directly | neg-Q gives a more concrete hypothesis | | 3 | Contradiction | Non-existence, irrationality, infinity | Negation yields a concrete object that self-destructs | | 4 | Cases | Natural partition of hypotheses | Parity, sign, size, set membership | | 5 | WLOG | Symmetric variables | Cases would repeat the same argument | | 6 | Vacuous/Trivial | Empty-set or always-true conclusio
- The Proof Toolbox at a Glance
- Technique 1 — Direct Proof
- Technique 2 — Proof by Contrapositive
- Technique 3 — Proof by Contradiction (Reductio ad Absurdum)
- Technique 4 — Proof by Cases
- Technique 5 — Without Loss of Generality (WLOG)
- Technique 6 — Vacuous and Trivial Proofs
- Technique 7 — Biconditional Proof
- Technique 8 — Counterexample (Disproof)
- Technique 9 — Existence Proofs
- 9a — Constructive Existence
- 9b — Non-Constructive Existence
- Technique 10 — Uniqueness Proofs
- Technique 11 — Induction
What does the proof-techniques skill do?
Proof writing and verification techniques for mathematical reasoning. Covers 13 proof strategies with canonical worked examples — direct, contrapositive, contradiction, cases, WLOG, vacuous/trivial, biconditional, counterexample, existence (constructive and non-constructive), uniqueness, induction (weak, strong, structural), pigeonhole/double-counting, and diagonal argument/invariants. Use when writing, verifying, or selecting proof strategies for any mathematical claim.
How do I install it?
Run `npx skills add majiayu000/claude-skill-registry --skill proof-techniques --agent claude-code` — it drops the skill into your project so the agent can pick it up. Swap the --agent value for codex, cursor or copilot if you use one of those.
Where does this skill come from?
From majiayu000/claude-skill-registry, a repository with 534 stars. We read it straight from the repository tree rather than a submitted listing, so what you see here is what is actually published.
Is a popular skill a good skill?
Not necessarily. Stars measure attention, not adoption — a repository can trend for a week and be abandoned. That is why we show the weekly change from our own snapshots next to the total, instead of a single flattering number.