/math-help

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 + 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/