Math Help

by parcadeid07ff4b06b62No license3.9K starsListed Oct 8, 2026Updated Oct 8, 2026Repository updated 8 months ago

Guide to the math cognitive stack - what tools exist and when to use each

Instructions onlyLearning & Education
AI-generated overview

A guide to a math tool stack, explaining which computation, proof, verification, tutoring or plotting tool to use.

What it does
This skill is a reference guide to a set of mathematical computation tools, organized into layers: symbolic algebra, constraint solving and theorem proving, reasoning verification, tutoring, and formal proofs. It lists commands and example invocations for each tool and provides decision flowcharts and common workflows. It produces guidance and command examples rather than computations itself.
When to use it
Use it when you need to decide which math tool fits a task, such as solving equations, checking a proof step, generating practice problems, or plotting functions. It is also useful for looking up example commands for the described tools.
Requirements
The guide itself is instructions only and ships no scripts. The tools it describes require a Python environment with uv, plus sympy, z3-solver, numpy, scipy, mpmath, matplotlib and plotly; Lean 4 is referenced through a separate skill.

Math Cognitive Stack Guide

Cognitive prosthetics for exact mathematical computation. This guide helps you choose the right tool for your math task.

Quick Reference

I want to...Use thisExample
Solve equationssympy_compute.py solvesolve "x**2 - 4 = 0" --var x
Integrate/differentiatesympy_compute.pyintegrate "sin(x)" --var x
Compute limitssympy_compute.py limitlimit "sin(x)/x" --var x --to 0
Matrix operationssympy_compute.py / numpy_compute.pydet "[[1,2],[3,4]]"
Verify a reasoning stepmath_scratchpad.py verifyverify "x = 2 implies x^2 = 4"
Check a proof chainmath_scratchpad.py chainchain --steps '[...]'
Get progressive hintsmath_tutor.py hinthint "Solve x^2 - 4 = 0" --level 2
Generate practice problemsmath_tutor.py generategenerate --topic algebra --difficulty 2
Prove a theorem (constraints)z3_solve.py proveprove "x + y == y + x" --vars x y
Check satisfiabilityz3_solve.py satsat "x > 0, x < 10, x*x == 49"
Optimize with constraintsz3_solve.py optimizeoptimize "x + y" --constraints "..."
Plot 2D/3D functionsmath_plot.pyplot2d "sin(x)" --range -10 10
Arbitrary precisionmpmath_compute.pypi --dps 100
Numerical optimizationscipy_compute.pyminimize "x**2 + 2*x" "5"
Formal machine proofLean 4 (lean4 skill)/lean4

The Five Layers

Layer 1: SymPy (Symbolic Algebra)

When: Exact algebraic computation - solving, calculus, simplification, matrix algebra.

Key Commands:

bash
# Solve equationuv run python -m runtime.harness scripts/sympy_compute.py \    solve "x**2 - 5*x + 6 = 0" --var x --domain real
# Integrateuv run python -m runtime.harness scripts/sympy_compute.py \    integrate "sin(x)" --var x
# Definite integraluv run python -m runtime.harness scripts/sympy_compute.py \    integrate "x**2" --var x --bounds 0 1
# Differentiate (2nd order)uv run python -m runtime.harness scripts/sympy_compute.py \    diff "x**3" --var x --order 2
# Simplify (trig strategy)uv run python -m runtime.harness scripts/sympy_compute.py \    simplify "sin(x)**2 + cos(x)**2" --strategy trig
# Limituv run python -m runtime.harness scripts/sympy_compute.py \    limit "sin(x)/x" --var x --to 0
# Matrix eigenvaluesuv run python -m runtime.harness scripts/sympy_compute.py \    eigenvalues "[[1,2],[3,4]]"

Best For: Closed-form solutions, calculus, exact algebra.

Layer 2: Z3 (Constraint Solving & Theorem Proving)

When: Proving theorems, checking satisfiability, constraint optimization.

Key Commands:

bash
# Prove commutativityuv run python -m runtime.harness scripts/cc_math/z3_solve.py \    prove "x + y == y + x" --vars x y --type int
# Check satisfiabilityuv run python -m runtime.harness scripts/cc_math/z3_solve.py \    sat "x > 0, x < 10, x*x == 49" --type int
# Optimizeuv run python -m runtime.harness scripts/cc_math/z3_solve.py \    optimize "x + y" --constraints "x >= 0, y >= 0, x + y <= 100" \    --direction maximize --type real

Best For: Logical proofs, constraint satisfaction, optimization with constraints.

Layer 3: Math Scratchpad (Reasoning Verification)

When: Verifying step-by-step reasoning, checking derivation chains.

Key Commands:

bash
# Verify single stepuv run python -m runtime.harness scripts/cc_math/math_scratchpad.py \    verify "x = 2 implies x^2 = 4"
# Verify with contextuv run python -m runtime.harness scripts/cc_math/math_scratchpad.py \    verify "x^2 = 4" --context '{"x": 2}'
# Verify chain of reasoninguv run python -m runtime.harness scripts/cc_math/math_scratchpad.py \    chain --steps '["x^2 - 4 = 0", "(x-2)(x+2) = 0", "x = 2 or x = -2"]'
# Explain a stepuv run python -m runtime.harness scripts/cc_math/math_scratchpad.py \    explain "d/dx(x^3) = 3*x^2"

Best For: Checking your work, validating derivations, step-by-step verification.

Layer 4: Math Tutor (Educational)

When: Learning, getting hints, generating practice problems.

Key Commands:

bash
# Step-by-step solutionuv run python scripts/cc_math/math_tutor.py steps "x**2 - 5*x + 6 = 0" --operation solve
# Progressive hint (level 1-5)uv run python scripts/cc_math/math_tutor.py hint "Solve x**2 - 4 = 0" --level 2
# Generate practice problemuv run python scripts/cc_math/math_tutor.py generate --topic algebra --difficulty 2

Best For: Learning, tutoring, practice.

Layer 5: Lean 4 (Formal Proofs)

When: Rigorous machine-verified mathematical proofs, category theory, type theory.

Access: Use /lean4 skill for full documentation.

Best For: Publication-grade proofs, dependent types, category theory.

Numerical Tools

For numerical (not symbolic) computation:

NumPy (160 functions)

bash
# Matrix operationsuv run python scripts/cc_math/numpy_compute.py det "[[1,2],[3,4]]"uv run python scripts/cc_math/numpy_compute.py inv "[[1,2],[3,4]]"uv run python scripts/cc_math/numpy_compute.py eig "[[1,2],[3,4]]"uv run python scripts/cc_math/numpy_compute.py svd "[[1,2,3],[4,5,6]]"
# Solve linear systemuv run python scripts/cc_math/numpy_compute.py solve "[[3,1],[1,2]]" "[9,8]"

SciPy (289 functions)

bash
# Minimize functionuv run python scripts/cc_math/scipy_compute.py minimize "x**2 + 2*x" "5"
# Find rootuv run python scripts/cc_math/scipy_compute.py root "x**3 - x - 2" "1.5"
# Curve fittinguv run python scripts/cc_math/scipy_compute.py curve_fit "a*exp(-b*x)" "0,1,2,3" "1,0.6,0.4,0.2" "1,0.5"

mpmath (153 functions, arbitrary precision)

bash
# Pi to 100 decimal placesuv run python scripts/cc_math/mpmath_compute.py pi --dps 100
# Arbitrary precision sqrtuv run python -m scripts.mpmath_compute mp_sqrt "2" --dps 100

Visualization

math_plot.py

bash
# 2D plotuv run python scripts/cc_math/math_plot.py plot2d "sin(x)" \    --var x --range -10 10 --output plot.png
# 3D surfaceuv run python scripts/cc_math/math_plot.py plot3d "x**2 + y**2" \    --xvar x --yvar y --range 5 --output surface.html
# Multiple functionsuv run python scripts/cc_math/math_plot.py plot2d-multi "sin(x),cos(x)" \    --var x --range -6.28 6.28 --output multi.png
# LaTeX renderinguv run python scripts/cc_math/math_plot.py latex "\\int e^{-x^2} dx" --output equation.png

Educational Features

5-Level Hint System

LevelCategoryWhat You Get
1ConceptualGeneral direction, topic identification
2StrategicApproach to use, technique selection
3TacticalSpecific steps, intermediate goals
4ComputationalIntermediate results, partial solutions
5AnswerFull solution with explanation

Usage:

bash
# Start with conceptual hintuv run python scripts/cc_math/math_tutor.py hint "integrate x*sin(x)" --level 1
# Get more specific guidanceuv run python scripts/cc_math/math_tutor.py hint "integrate x*sin(x)" --level 3

Step-by-Step Solutions

bash
uv run python scripts/cc_math/math_tutor.py steps "x**2 - 5*x + 6 = 0" --operation solve

Returns structured steps with:

  • Step number and type
  • From/to expressions
  • Rule applied
  • Justification

Common Workflows

Workflow 1: Solve and Verify

  1. Solve with sympy_compute.py
  2. Verify solution with math_scratchpad.py
  3. Plot to visualize (optional)
bash
# Solveuv run python -m runtime.harness scripts/sympy_compute.py \    solve "x**2 - 4 = 0" --var x
# Verify the solutions workuv run python -m runtime.harness scripts/cc_math/math_scratchpad.py \    verify "x = 2 implies x^2 - 4 = 0"

Workflow 2: Learn a Concept

  1. Generate practice problem with math_tutor.py
  2. Use progressive hints (level 1, then 2, etc.)
  3. Get full solution if stuck
bash
# Generate problemuv run python scripts/cc_math/math_tutor.py generate --topic calculus --difficulty 2
# Get hints progressivelyuv run python scripts/cc_math/math_tutor.py hint "..." --level 1uv run python scripts/cc_math/math_tutor.py hint "..." --level 2
# Full solutionuv run python scripts/cc_math/math_tutor.py steps "..." --operation integrate

Workflow 3: Prove and Formalize

  1. Check theorem with z3_solve.py (constraint-level proof)
  2. If rigorous proof needed, use Lean 4
bash
# Quick check with Z3uv run python -m runtime.harness scripts/cc_math/z3_solve.py \    prove "x*y == y*x" --vars x y --type int
# For formal proof, use /lean4 skill

Choosing the Right Tool

Is it SYMBOLIC (exact answers)?  └─ Yes → Use SymPy      ├─ Equations → sympy_compute.py solve      ├─ Calculus → sympy_compute.py integrate/diff/limit      └─ Simplify → sympy_compute.py simplify
Is it a PROOF or CONSTRAINT problem?  └─ Yes → Use Z3      ├─ True/False theorem → z3_solve.py prove      ├─ Find values → z3_solve.py sat      └─ Optimize → z3_solve.py optimize
Is it NUMERICAL (approximate answers)?  └─ Yes → Use NumPy/SciPy      ├─ Linear algebra → numpy_compute.py      ├─ Optimization → scipy_compute.py minimize      └─ High precision → mpmath_compute.py
Need to VERIFY reasoning?  └─ Yes → Use Math Scratchpad      ├─ Single step → math_scratchpad.py verify      └─ Chain → math_scratchpad.py chain
Want to LEARN/PRACTICE?  └─ Yes → Use Math Tutor      ├─ Hints → math_tutor.py hint      └─ Practice → math_tutor.py generate
Need MACHINE-VERIFIED formal proof?  └─ Yes → Use Lean 4 (see /lean4 skill)

Related Skills

  • /math or /math-mode - Quick access to the orchestration skill
  • /lean4 - Formal theorem proving with Lean 4
  • /lean4-functors - Category theory functors
  • /lean4-nat-trans - Natural transformations
  • /lean4-limits - Limits and colimits

Requirements

All math scripts are installed via:

bash
uv sync

Dependencies: sympy, z3-solver, numpy, scipy, mpmath, matplotlib, plotly

Source and attribution

Source:parcadei/continuous-claude-v3in.claude/skills/math-helpat commitd07ff4b

License: No license

Content belongs to its original authors. SourceWeft indexes it from a public repository.

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