Enter your mathematical statement and the steps used to prove it. This tool will attempt a basic verification of the logical flow.
Verification Result:
Understanding the Math Proof Verifier
This tool is designed to assist in understanding and performing a basic verification of mathematical proofs. Mathematical proofs are rigorous logical arguments that establish the truth of a mathematical statement. They are the cornerstone of mathematics, providing certainty and a foundation for further exploration.
What is a Mathematical Proof?
A mathematical proof is a sequence of logical deductions, starting from axioms (statements assumed to be true) or previously proven theorems, that leads to a specific conclusion. Each step in a proof must be justified by accepted mathematical principles, definitions, or previously established results.
How This Verifier Works (Simplified)
The 'Math Proof Verifier' takes your mathematical statement and the steps you've outlined for its proof. It then performs a rudimentary check by:
Analyzing Structure: It looks for common proof structures like direct proofs, proofs by contradiction, or proofs by induction (though complex induction might not be fully parsed).
Identifying Logical Connectors: It tries to recognize keywords and phrases that indicate logical progression (e.g., "If… then…", "Therefore…", "Since…", "Consider cases…").
Basic Step Coherence: It checks if subsequent steps generally build upon previous ones.
Important Limitation: This verifier is a tool for structural and basic logical flow analysis. It does not perform deep symbolic computation or algebraic manipulation. It cannot definitively 'prove' a statement is true in the same way a human mathematician or specialized theorem prover can. Its purpose is to highlight potential gaps or inconsistencies in the *presentation* of a proof, encouraging users to refine their arguments.
Use Cases:
Students: Use it to check if their written proofs have a coherent structure and logical flow before submitting them to instructors.
Educators: Demonstrate the components of a proof and how this tool can offer initial feedback.
Learners: Explore examples of proofs and see how different statements and steps are interpreted.
Example Scenario:
Statement: For any integer $n$, the expression $n^2 + n$ is always an even number.
Proof Steps Provided:
Let $n$ be any integer.
We can consider two cases for $n$: $n$ is even or $n$ is odd.
Case 1: $n$ is even. If $n$ is even, we can write $n = 2k$ for some integer $k$. Substituting this into the expression: $n^2 + n = (2k)^2 + (2k) = 4k^2 + 2k = 2(2k^2 + k)$. Since $2k^2 + k$ is an integer, the result is of the form $2 \times (\text{integer})$, which means $n^2 + n$ is even.
Case 2: $n$ is odd. If $n$ is odd, we can write $n = 2k + 1$ for some integer $k$. Substituting this: $n^2 + n = (2k+1)^2 + (2k+1) = (4k^2 + 4k + 1) + (2k + 1) = 4k^2 + 6k + 2 = 2(2k^2 + 3k + 1)$. Since $2k^2 + 3k + 1$ is an integer, the result is of the form $2 \times (\text{integer})$, meaning $n^2 + n$ is also even in this case.
Since $n^2 + n$ is even for both even and odd integers $n$, the statement is proven true for all integers.
The verifier would analyze these steps, recognize the case analysis, the substitution, and the conclusion, likely deeming the structure sound (while not performing the algebraic checks itself).
function verifyProof() {
var statement = document.getElementById("statement").value.trim();
var proofStepsText = document.getElementById("proofSteps").value.trim();
var proofResultDiv = document.getElementById("proofResult");
var errorResultDiv = document.getElementById("errorResult");
// Clear previous results
proofResultDiv.innerHTML = "";
errorResultDiv.innerHTML = "";
if (!statement || !proofStepsText) {
errorResultDiv.innerHTML = "Please enter both the mathematical statement and the proof steps.";
return;
}
var steps = proofStepsText.split('\n').map(function(step) {
return step.trim();
}).filter(function(step) {
return step.length > 0; // Remove empty lines
});
if (steps.length < 2) {
errorResultDiv.innerHTML = "Proof requires at least two steps for verification.";
return;
}
// — Basic Verification Logic —
// This is a highly simplified "verifier". Real proof verification is extremely complex.
// This checks for some common structural elements and logical connectors.
var hasIntroduction = false;
var hasConclusion = false;
var hasLogicalConnectors = 0;
var potentialIssues = [];
// Check for common introductory phrases
var introKeywords = ["let", "assume", "suppose", "consider"];
var statementLower = statement.toLowerCase();
for (var i = 0; i < introKeywords.length; i++) {
if (statementLower.includes(introKeywords[i]) || steps[0].toLowerCase().includes(introKeywords[i])) {
hasIntroduction = true;
break;
}
}
// Check for common concluding phrases
var conclusionKeywords = ["therefore", "hence", "thus", "conclude", "proven", "it follows that"];
for (var i = 0; i < steps.length; i++) {
var stepLower = steps[i].toLowerCase();
if (stepLower.includes("proven") || stepLower.includes("statement is true")) {
hasConclusion = true;
break;
}
// Check last step more rigorously for conclusion keywords
if (i === steps.length – 1) {
for (var j = 0; j < conclusionKeywords.length; j++) {
if (stepLower.includes(conclusionKeywords[j])) {
hasConclusion = true;
break;
}
}
}
}
// Check for logical connectors within steps
var connectorKeywords = ["if", "then", "since", "because", "for", "case", "also", "similarly", "however", "but"];
for (var i = 0; i < steps.length; i++) {
var stepLower = steps[i].toLowerCase();
for (var j = 0; j steps.length / 2) overallScore++; // More than half the steps have connectors
// Basic rule checks (very crude)
if (!hasIntroduction) {
potentialIssues.push("Proof lacks a clear starting point or introduction.");
}
if (!hasConclusion) {
potentialIssues.push("Proof does not seem to have a clear concluding statement linking back to the original statement.");
}
if (hasLogicalConnectors 3) {
potentialIssues.push("Proof appears to have limited logical connectors between steps.");
}
if (steps.some(step => step.length > 150)) {
potentialIssues.push("Some steps are very long, consider breaking them down for clarity.");
}
// Determine the result message
if (overallScore >= 2 && potentialIssues.length === 0) {
proofResultDiv.innerHTML = "Looks good! The proof structure seems logical and well-formed.";
} else if (overallScore >= 1 && potentialIssues.length < 3) {
proofResultDiv.innerHTML = "Potentially valid. Structure is somewhat logical, but review suggested issues.";
errorResultDiv.innerHTML = "Review needed: " + potentialIssues.join(" | ");
}
else {
proofResultDiv.innerHTML = "Structural concerns detected. Please review the steps carefully.";
errorResultDiv.innerHTML = "Issues found: " + potentialIssues.join(" | ");
}
}