Automatically Verify Estimates in Analysis with Python: A Practical Guide

Are you tired of manually checking estimates in your analysis? Discover how to automate the process of proving or verifying bounds using Python. This guide provides a straightforward approach to ensure the accuracy of your calculations, saving you time and reducing the risk of errors.

Simplify Bound Verification with Automated Case Splitting

This code focuses on proving bounds involving positive quantities by utilizing brute-force case splitting. This might not be the most efficient method, but it provides a reliable way to verify your estimates.

• Benefit: Ensures estimate accuracy.
• Benefit: Reduces errors in manual calculations.

Proving Inequalities: A Hands-On Example

Let's start with a simple example: proving the relationship between the minimum and maximum of two positive variables, a and b. Here's how you can do it:

a = Variable("a")
b = Variable("b")
assumptions = Assumptions()
assumptions.can_bound(min(a, b), max(a, b))

The code automatically splits the problem into different cases (e.g., a < b, b < a) and attempts to prove the bound in each case.

• Benefit: Simplifies complex inequality proofs.
• Benefit: Automatically explores different scenarios.
• Benefit: Provides detailed verification output.

Weak Form of AM-GM Inequality with Python

Verifying mathematical inequalities such as the AM-GM inequality can be cumbersome. Automate verification using the following code snippet:

a = Variable("a")
b = Variable("b")
c = Variable("c")
assumptions = Assumptions()
assumptions.can_bound((a * b * c) ** (1 / 3), max(a, b, c))

The script will verify if the cube root of (a * b * c) is bounded by the maximum of a, b, and c.

• Benefit: Checks mathematical assumptions.
• Benefit: Catches errors early on.

Adding Initial Hypotheses to Refine Proofs

Sometimes, you need to add initial assumptions to guide the verification process. For example, let's say you want to prove that if a <= b, b <= c, and c <= d, then a * c <= b * d. Here's how you can incorporate these hypotheses:

a = Variable("a")
b = Variable("b")
c = Variable("c")
d = Variable("d")
assumptions = Assumptions()
assumptions.add(a <= b)
assumptions.add(b <= c)
assumptions.add(c <= d)
assumptions.can_bound(a * c, b * d)

By adding these assumptions, you provide the code with the necessary context to prove the desired bound.

• Benefit: Allows precise and focused analysis.
• Benefit: Guides the verification towards a specific conclusion.

Recognizing Limitations: When Hypotheses Aren't Enough

It's important to recognize when the given hypotheses are insufficient to prove a claim. The script will indicate when verification is not possible. This helps you identify gaps in your reasoning or missing assumptions.

a = Variable("a")
b = Variable("b")
c = Variable("c")
d = Variable("d")
assumptions = Assumptions()
assumptions.add(a <= b)
assumptions.add(b <= c)
assumptions.add(c <= d)
assumptions.can_bound(a * d, b * c)

• Benefit: Avoids wasting time on unprovable statements.
• Benefit: Identifies the need for additional information or assumptions.

Incorporating Littlewood-Paley Conditions

Littlewood-Paley conditions are important in areas like PDE analysis to analyze nonlinear interactions between different frequencies. The following example provides the method to implement this:

a = Variable("a")
b = Variable("b")
c = Variable("c")
assumptions = Assumptions()
assumptions.add_lp(a,b,c)
assumptions.can_bound(max(a, b, c)**2 * min(a,b,c), a * b * c)

• Benefit: Broadens applicability to advanced mathematical concepts.
• Benefit: Applicable in PDE (Partial Differential Equations).

Get Started with Automated Verification

By using this code, you can significantly improve the rigor and efficiency of your analysis. Whether you're working with simple inequalities or complex mathematical conditions, this automation tool can help you verify your estimates and gain confidence in your results.