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/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