name	math-help
description	Guide to the math cognitive stack - what tools exist and when to use each
triggers	["help","guide","how do I","what math","math help","math tools","which tool","math tutorial"]
user-invocable	false

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 this	Example
Solve equations	sympy_compute.py solve	`solve "x**2 - 4 = 0" --var x`
Integrate/differentiate	sympy_compute.py	`integrate "sin(x)" --var x`
Compute limits	sympy_compute.py limit	`limit "sin(x)/x" --var x --to 0`
Matrix operations	sympy_compute.py / numpy_compute.py	`det "[[1,2],[3,4]]"`
Verify a reasoning step	math_scratchpad.py verify	`verify "x = 2 implies x^2 = 4"`
Check a proof chain	math_scratchpad.py chain	`chain --steps '[...]'`
Get progressive hints	math_tutor.py hint	`hint "Solve x^2 - 4 = 0" --level 2`
Generate practice problems	math_tutor.py generate	`generate --topic algebra --difficulty 2`
Prove a theorem (constraints)	z3_solve.py prove	`prove "x + y == y + x" --vars x y`
Check satisfiability	z3_solve.py sat	`sat "x > 0, x < 10, x*x == 49"`
Optimize with constraints	z3_solve.py optimize	`optimize "x + y" --constraints "..."`
Plot 2D/3D functions	math_plot.py	`plot2d "sin(x)" --range -10 10`
Arbitrary precision	mpmath_compute.py	`pi --dps 100`
Numerical optimization	scipy_compute.py	`minimize "x*2 + 2x" "5"`
Formal machine proof	Lean 4 (lean4 skill)	`/lean4`

The Five Layers

Layer 1: SymPy (Symbolic Algebra)

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

Key Commands:

# Solve equation
uv run python -m runtime.harness scripts/sympy_compute.py \
    solve "x**2 - 5*x + 6 = 0" --var x --domain real

# Integrate
uv run python -m runtime.harness scripts/sympy_compute.py \
    integrate "sin(x)" --var x

# Definite integral
uv 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

# Limit
uv run python -m runtime.harness scripts/sympy_compute.py \
    limit "sin(x)/x" --var x --to 0

# Matrix eigenvalues
uv 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:

# Prove commutativity
uv run python -m runtime.harness scripts/cc_math/z3_solve.py \
    prove "x + y == y + x" --vars x y --type int

# Check satisfiability
uv run python -m runtime.harness scripts/cc_math/z3_solve.py \
    sat "x > 0, x < 10, x*x == 49" --type int

# Optimize
uv 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:

# Verify single step
uv run python -m runtime.harness scripts/cc_math/math_scratchpad.py \
    verify "x = 2 implies x^2 = 4"

# Verify with context
uv run python -m runtime.harness scripts/cc_math/math_scratchpad.py \
    verify "x^2 = 4" --context '{"x": 2}'

# Verify chain of reasoning
uv 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 step
uv 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:

# Step-by-step solution
uv 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 problem
uv 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)

# Matrix operations
uv 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 system
uv run python scripts/cc_math/numpy_compute.py solve "[[3,1],[1,2]]" "[9,8]"

SciPy (289 functions)

# Minimize function
uv run python scripts/cc_math/scipy_compute.py minimize "x**2 + 2*x" "5"

# Find root
uv run python scripts/cc_math/scipy_compute.py root "x**3 - x - 2" "1.5"

# Curve fitting
uv 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)

# Pi to 100 decimal places
uv run python scripts/cc_math/mpmath_compute.py pi --dps 100

# Arbitrary precision sqrt
uv run python -m scripts.mpmath_compute mp_sqrt "2" --dps 100

Visualization

math_plot.py

# 2D plot
uv run python scripts/cc_math/math_plot.py plot2d "sin(x)" \
    --var x --range -10 10 --output plot.png

# 3D surface
uv run python scripts/cc_math/math_plot.py plot3d "x**2 + y**2" \
    --xvar x --yvar y --range 5 --output surface.html

# Multiple functions
uv 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 rendering
uv run python scripts/cc_math/math_plot.py latex "\\int e^{-x^2} dx" --output equation.png

Educational Features

5-Level Hint System

Level	Category	What You Get
1	Conceptual	General direction, topic identification
2	Strategic	Approach to use, technique selection
3	Tactical	Specific steps, intermediate goals
4	Computational	Intermediate results, partial solutions
5	Answer	Full solution with explanation

Usage:

# Start with conceptual hint
uv run python scripts/cc_math/math_tutor.py hint "integrate x*sin(x)" --level 1

# Get more specific guidance
uv run python scripts/cc_math/math_tutor.py hint "integrate x*sin(x)" --level 3

Step-by-Step Solutions

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

Solve with sympy_compute.py
Verify solution with math_scratchpad.py
Plot to visualize (optional)

# Solve
uv run python -m runtime.harness scripts/sympy_compute.py \
    solve "x**2 - 4 = 0" --var x

# Verify the solutions work
uv run python -m runtime.harness scripts/cc_math/math_scratchpad.py \
    verify "x = 2 implies x^2 - 4 = 0"

Workflow 2: Learn a Concept

Generate practice problem with math_tutor.py
Use progressive hints (level 1, then 2, etc.)
Get full solution if stuck

# Generate problem
uv run python scripts/cc_math/math_tutor.py generate --topic calculus --difficulty 2

# Get hints progressively
uv run python scripts/cc_math/math_tutor.py hint "..." --level 1
uv run python scripts/cc_math/math_tutor.py hint "..." --level 2

# Full solution
uv run python scripts/cc_math/math_tutor.py steps "..." --operation integrate

Workflow 3: Prove and Formalize

Check theorem with z3_solve.py (constraint-level proof)
If rigorous proof needed, use Lean 4

# Quick check with Z3
uv 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:

uv sync

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