Math Reasoning

Mathematical Reasoning

Perform rigorous mathematical reasoning and produce publication-quality LaTeX output.

Input

• $0 — Task type: derive, prove, formalize, stats, notation, verify
• $1 — Context: equation, theorem statement, problem description, or data description

Tasks

derive — Step-by-step equation derivation

Show every intermediate step. Justify each with the rule applied. Box final result with \boxed{}. Number important equations with \label{eq:name}.

prove — Formal theorem proof

Use appropriate technique: direct, contradiction, induction, construction, or cases. See references/proof-templates.md for LaTeX templates.

formalize — Problem setting formalization

Convert informal description into formal mathematical framework with: variable definitions, domain/range specifications, assumptions, objective function.

stats — Statistical test selection

Use the decision tree in references/notation-guide.md to select appropriate tests. Report p-values, effect sizes, confidence intervals.

notation — Generate notation table

Create a \begin{table} with all symbols used in the paper. Use standard ML notation from references/notation-guide.md.

verify — Check mathematical correctness

Verify: dimensional consistency, boundary cases, gradient computations, notation consistency across sections.

References

• Standard ML notation + statistical tests: ~/.claude/skills/math-reasoning/references/notation-guide.md
• Proof templates and theorem environments: ~/.claude/skills/math-reasoning/references/proof-templates.md

Rules

• Define ALL symbols before first use: "Let $\mathcal{X}$ denote..."
• Use consistent notation throughout the paper
• Number equations that are referenced later
• Use \tag{reason} for key derivation steps
• State assumptions explicitly
• Cite lemmas and prior results used in proofs

Related Skills

• Upstream: research-planning
• Downstream: algorithm-design, paper-writing-section
• See also: symbolic-equation, data-analysis

Standard Mathematical Notation for ML/AI Papers

Spaces and Sets

Probability and Statistics

Optimization

Linear Algebra

Common ML Operations

Useful LaTeX Declarations

\DeclareMathOperator*{\argmin}{arg\,min}
\DeclareMathOperator*{\argmax}{arg\,max}
\DeclareMathOperator{\tr}{tr}
\DeclareMathOperator{\diag}{diag}
\DeclareMathOperator{\rank}{rank}
\DeclareMathOperator{\softmax}{softmax}
\DeclareMathOperator{\sigmoid}{\sigma}

\newcommand{\R}{\mathbb{R}}
\newcommand{\E}{\mathbb{E}}
\newcommand{\Prob}{\mathbb{P}}
\newcommand{\bx}{\mathbf{x}}
\newcommand{\by}{\mathbf{y}}
\newcommand{\bW}{\mathbf{W}}
\newcommand{\btheta}{\boldsymbol{\theta}}
\newcommand{\norm}[1]{\left\| #1 \right\|}
\newcommand{\abs}[1]{\left| #1 \right|}
\newcommand{\inner}[2]{\langle #1, #2 \rangle}
\newcommand{\KL}[2]{D_{\mathrm{KL}}\left( #1 \| #2 \right)}

Statistical Tests Decision Tree

Comparing two groups

• Normal data → paired/unpaired t-test
• Non-normal → Wilcoxon signed-rank / Mann-Whitney U

Comparing >2 groups

• Normal + equal variance → one-way ANOVA + post-hoc Tukey HSD
• Otherwise → Kruskal-Wallis + post-hoc Dunn

Correlation

• Linear → Pearson's r
• Monotonic → Spearman's ρ

Categorical data

• 2×2 → Fisher's exact test / χ² test
• Larger → χ² test of independence

Multiple comparisons

• Always apply Bonferroni correction or Benjamini-Hochberg FDR
• Report: α_corrected = α / k where k = number of comparisons

Reporting convention

The improvement is statistically significant
($p < 0.01$, paired $t$-test, $t(4) = 5.23$, Cohen's $d = 1.87$).
Results reported as mean $\pm$ std over $N = 5$ random seeds.

Proof Templates for Research Papers

Theorem Environment

\begin{theorem}[Name]
\label{thm:name}
Let $f: \mathcal{X} \to \mathcal{Y}$ be a continuous function. Then ...
\end{theorem}

\begin{proof}
We prove by [technique]. ...

[Step 1] ...
[Step 2] ...
Therefore, ... \qed
\end{proof}

Supporting Environments

\begin{lemma}[Helper result]
\label{lem:helper}
For all $x \in \mathcal{X}$, ...
\end{lemma}

\begin{proposition}
\label{prop:name}
Under Assumption~\ref{asm:name}, ...
\end{proposition}

\begin{corollary}
\label{cor:name}
As a direct consequence of Theorem~\ref{thm:name}, ...
\end{corollary}

\begin{definition}[Concept Name]
\label{def:name}
We define ... as ...
\end{definition}

\begin{assumption}
\label{asm:name}
We assume that ...
\end{assumption}

\begin{remark}
Note that this result implies ...
\end{remark}

Proof Techniques

Direct Proof

\begin{proof}
Assume the premises hold. We have:
\begin{align}
    f(x) &= ... \label{eq:step1} \\
    &\leq ... \tag{by Lemma~\ref{lem:helper}} \\
    &= ... \nonumber
\end{align}
which completes the proof.
\end{proof}

Proof by Contradiction

\begin{proof}
Suppose for contradiction that $\neg P$. Then ...
This contradicts the assumption that ..., completing the proof.
\end{proof}

Proof by Induction

\begin{proof}
We prove by induction on $n$.

\textbf{Base case} ($n = 1$): ...

\textbf{Inductive step}: Assume the statement holds for $n = k$.
We show it holds for $n = k + 1$:
\begin{align}
    f(k+1) &= f(k) + g(k) \\
    &\leq ... \tag{by inductive hypothesis} \\
    &= ...
\end{align}
By the principle of mathematical induction, the result holds for all $n \geq 1$.
\end{proof}

Convergence Proof (common in ML)

\begin{theorem}[Convergence Rate]
\label{thm:convergence}
Under Assumptions~\ref{asm:smoothness} and~\ref{asm:bounded_var}, Algorithm~\ref{alg:method}
with step size $\eta = \frac{1}{\sqrt{T}}$ satisfies:
\begin{equation}
    \frac{1}{T} \sum_{t=1}^{T} \mathbb{E}\left[\|\nabla f(\theta_t)\|^2\right] \leq \frac{2(f(\theta_1) - f^*)}{\sqrt{T}} + \frac{L\sigma^2}{\sqrt{T}}
\end{equation}
\end{theorem}

Generalization Bound (common in learning theory)

\begin{theorem}[Generalization Bound]
\label{thm:generalization}
Let $\mathcal{F}$ be a function class with Rademacher complexity $\mathfrak{R}_n(\mathcal{F})$.
For any $\delta > 0$, with probability at least $1 - \delta$ over the draw of $n$ samples:
\begin{equation}
    \sup_{f \in \mathcal{F}} \left| \hat{R}(f) - R(f) \right| \leq 2\mathfrak{R}_n(\mathcal{F}) + \sqrt{\frac{\log(2/\delta)}{2n}}
\end{equation}
\end{theorem}

Complexity Analysis Template

\begin{proposition}[Computational Complexity]
\label{prop:complexity}
Algorithm~\ref{alg:method} has time complexity $\mathcal{O}(nd^2)$ and
space complexity $\mathcal{O}(nd)$, where $n$ is the number of samples
and $d$ is the input dimension.
\end{proposition}