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 + 2*x" "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