#![allow(clippy::all)]
use scirs2_core::ndarray::{Array1, Array2};
use scirs2_core::Complex64;
use scirs2_special::*;
use std::collections::HashMap;
use std::f64::consts::PI;
use std::io::{self, Write};
use std::time::{Duration, Instant};
#[allow(dead_code)]
#[derive(Debug, Clone)]
struct ProofExplorer {
available_proofs: Vec<ProofModule>,
user_progress: UserProgress,
current_proof: Option<ProofSession>,
difficulty_settings: DifficultySettings,
}
#[allow(dead_code)]
#[derive(Debug, Clone)]
struct ProofModule {
id: String,
title: String,
description: String,
mathematical_level: MathematicalLevel,
prerequisites: Vec<String>,
main_theorem: Theorem,
proof_steps: Vec<ProofStep>,
applications: Vec<Application>,
related_proofs: Vec<String>,
computational_examples: Vec<ComputationalExample>,
}
#[allow(dead_code)]
#[derive(Debug, Clone)]
enum MathematicalLevel {
Undergraduate,
Graduate,
Advanced,
Research,
}
#[allow(dead_code)]
#[derive(Debug, Clone)]
struct Theorem {
statement: String,
mathematical_formulation: String,
conditions: Vec<String>,
significance: String,
historical_context: String,
}
#[allow(dead_code)]
#[derive(Debug, Clone)]
struct ProofStep {
id: String,
title: String,
explanation: String,
mathematical_content: String,
justification: String,
interactive_elements: Vec<InteractiveElement>,
verification_method: Option<VerificationMethod>,
common_confusions: Vec<String>,
hints: Vec<String>,
}
#[allow(dead_code)]
#[derive(Debug, Clone)]
enum InteractiveElement {
UserInput {
prompt: String,
expected_answer: String,
validation_tolerance: f64,
hints: Vec<String>,
},
Visualization {
title: String,
description: String,
plot_type: PlotType,
},
Computation {
description: String,
code_example: String,
expected_result: String,
},
ConceptualQuestion {
question: String,
options: Vec<String>,
correct_answer: usize,
explanation: String,
},
}
#[allow(dead_code)]
#[derive(Debug, Clone)]
enum PlotType {
FunctionGraph {
domain: (f64, f64),
functions: Vec<String>,
},
ComplexPlane {
radius: f64,
function: String,
},
Convergence {
series: String,
terms: usize,
},
AsymptoticComparison {
exact: String,
approximation: String,
domain: (f64, f64),
},
}
#[allow(dead_code)]
#[derive(Debug, Clone)]
enum VerificationMethod {
NumericalCheck {
description: String,
test_cases: Vec<TestCase>,
},
SymbolicVerification {
description: String,
identity: String,
},
AsymptoticVerification {
description: String,
limit_expression: String,
},
}
#[allow(dead_code)]
#[derive(Debug, Clone)]
struct TestCase {
input: f64,
expected: f64,
tolerance: f64,
description: String,
}
#[allow(dead_code)]
#[derive(Debug, Clone)]
struct Application {
title: String,
description: String,
mathematical_context: String,
practical_example: String,
}
#[allow(dead_code)]
#[derive(Debug, Clone)]
struct ComputationalExample {
title: String,
description: String,
code: String,
expected_output: String,
learning_objective: String,
}
#[allow(dead_code)]
#[derive(Debug, Clone)]
struct UserProgress {
completed_proofs: Vec<String>,
difficulty_level: MathematicalLevel,
understanding_scores: HashMap<String, f64>,
time_spent: HashMap<String, Duration>,
favorite_topics: Vec<String>,
achievements: Vec<String>,
}
#[allow(dead_code)]
#[derive(Debug, Clone)]
struct ProofSession {
proof_id: String,
current_step: usize,
start_time: Instant,
step_completion_times: Vec<Duration>,
user_responses: Vec<UserResponse>,
understanding_score: f64,
}
#[allow(dead_code)]
#[derive(Debug, Clone)]
struct UserResponse {
step_id: String,
response: String,
correct: bool,
attempts: u32,
time_taken: Duration,
}
#[allow(dead_code)]
#[derive(Debug, Clone)]
struct DifficultySettings {
show_hints_automatically: bool,
require_step_verification: bool,
include_computational_checks: bool,
mathematical_rigor_level: MathematicalLevel,
}
impl ProofExplorer {
fn new() -> Self {
let mut explorer = Self {
available_proofs: Vec::new(),
user_progress: UserProgress {
completed_proofs: Vec::new(),
difficulty_level: MathematicalLevel::Undergraduate,
understanding_scores: HashMap::new(),
time_spent: HashMap::new(),
favorite_topics: Vec::new(),
achievements: Vec::new(),
},
current_proof: None,
difficulty_settings: DifficultySettings {
show_hints_automatically: true,
require_step_verification: true,
include_computational_checks: true,
mathematical_rigor_level: MathematicalLevel::Undergraduate,
},
};
explorer.initialize_proof_library();
explorer
}
fn initialize_proof_library(&mut self) {
self.available_proofs.push(ProofModule {
id: "gamma_reflection".to_string(),
title: "Gamma Function Reflection Formula".to_string(),
description: "A beautiful proof connecting the gamma function to trigonometry through complex analysis".to_string(),
mathematical_level: MathematicalLevel::Graduate,
prerequisites: vec![
"complex_analysis".to_string(),
"gamma_function_basics".to_string(),
"residue_calculus".to_string(),
],
main_theorem: Theorem {
statement: "For all z not a non-positive integer, Γ(z)Γ(1-z) = π/sin(πz)".to_string(),
mathematical_formulation: "\\Gamma(z)\\Gamma(1-z) = \\frac{\\pi}{\\sin(\\pi z)}".to_string(),
conditions: vec![
"z ∉ {..., -2, -1, 0}".to_string(),
"The formula extends by analytic continuation".to_string(),
],
significance: "This formula connects discrete factorials to continuous trigonometric functions, demonstrating the deep unity of mathematics".to_string(),
historical_context: "Discovered by Euler as part of his investigation into extending factorials to non-integer values".to_string(),
},
proof_steps: create_reflection_formula_proof_steps(),
applications: vec![
Application {
title: "Beta Function Simplification".to_string(),
description: "Simplifies beta function evaluations for special arguments".to_string(),
mathematical_context: "B(z, 1-z) = Γ(z)Γ(1-z) = π/sin(πz)".to_string(),
practical_example: "Computing B(1/3, 2/3) = 2π/√3".to_string(),
},
Application {
title: "Special Values of Gamma".to_string(),
description: "Provides a method to compute Γ(1/2), Γ(1/3), etc.".to_string(),
mathematical_context: "For z = 1/2: Γ(1/2)² = π".to_string(),
practical_example: "Γ(1/2) = √π ≈ 1.77245385".to_string(),
},
],
related_proofs: vec![
"gamma_duplication".to_string(),
"beta_function_properties".to_string(),
],
computational_examples: create_reflection_formula_examples(),
});
self.available_proofs.push(ProofModule {
id: "stirling_asymptotic".to_string(),
title: "Stirling's Asymptotic Series".to_string(),
description: "Derivation of the complete asymptotic expansion of the gamma function using the Euler-Maclaurin formula".to_string(),
mathematical_level: MathematicalLevel::Advanced,
prerequisites: vec![
"euler_maclaurin_formula".to_string(),
"asymptotic_series".to_string(),
"bernoulli_numbers".to_string(),
],
main_theorem: Theorem {
statement: "For large |z| with |arg z| < π: ln Γ(z) ~ (z-1/2)ln z - z + (1/2)ln(2π) + asymptotic series".to_string(),
mathematical_formulation: "\\ln \\Gamma(z) \\sim \\left(z - \\frac{1}{2}\\right) \\ln z - z + \\frac{1}{2}\\ln(2\\pi) + \\sum_{k=1}^{\\infty} \\frac{B_{2k}}{2k(2k-1)z^{2k-1}}".to_string(),
conditions: vec![
"|z| → ∞".to_string(),
"|arg z| < π".to_string(),
"Series is asymptotic, not convergent".to_string(),
],
significance: "Provides highly accurate approximations for large arguments and connects gamma function to Bernoulli numbers".to_string(),
historical_context: "Stirling's original approximation was refined by many mathematicians to give the complete asymptotic series".to_string(),
},
proof_steps: create_stirling_proof_steps(),
applications: vec![
Application {
title: "High-Precision Gamma Computation".to_string(),
description: "Enables accurate computation of gamma function for large arguments".to_string(),
mathematical_context: "Each term in the series provides additional precision".to_string(),
practical_example: "Computing Γ(100) with 15-digit accuracy".to_string(),
},
],
related_proofs: vec!["euler_maclaurin_derivation".to_string()],
computational_examples: create_stirling_examples(),
});
self.add_additional_proofs();
}
fn add_additional_proofs(&mut self) {
self.available_proofs.push(ProofModule {
id: "bessel_generating".to_string(),
title: "Bessel Function Generating Function".to_string(),
description: "Proof that exp(x(t-1/t)/2) generates the Bessel functions".to_string(),
mathematical_level: MathematicalLevel::Graduate,
prerequisites: vec![
"power_series".to_string(),
"bessel_functions".to_string(),
"complex_analysis".to_string(),
],
main_theorem: Theorem {
statement: "The exponential exp(x(t-1/t)/2) is the generating function for Bessel functions of the first kind".to_string(),
mathematical_formulation: "e^{\\frac{x}{2}(t-\\frac{1}{t})} = \\sum_{n=-\\infty}^{\\infty} J_n(x) t^n".to_string(),
conditions: vec![
"x ∈ ℂ".to_string(),
"t ≠ 0".to_string(),
"Series converges for all finite x and t ≠ 0".to_string(),
],
significance: "Provides a unified approach to deriving properties of all Bessel functions simultaneously".to_string(),
historical_context: "This generating function was crucial in Bessel's original work on planetary motion".to_string(),
},
proof_steps: create_bessel_generating_proof_steps(),
applications: vec![
Application {
title: "Bessel Function Identities".to_string(),
description: "Generates recurrence relations and addition formulas".to_string(),
mathematical_context: "Differentiating the generating function gives identities".to_string(),
practical_example: "Deriving J_{n-1}(x) + J_{n+1}(x) = (2n/x)J_n(x)".to_string(),
},
],
related_proofs: vec!["bessel_recurrence_relations".to_string()],
computational_examples: create_bessel_generating_examples(),
});
self.available_proofs.push(ProofModule {
id: "error_function_series".to_string(),
title: "Error Function Series Expansion".to_string(),
description: "Derivation of the power series for the error function through term-by-term integration".to_string(),
mathematical_level: MathematicalLevel::Undergraduate,
prerequisites: vec![
"power_series".to_string(),
"integration".to_string(),
"gaussian_integral".to_string(),
],
main_theorem: Theorem {
statement: "The error function has the series expansion erf(x) = (2/√π) Σ (-1)^n x^(2n+1) / (n!(2n+1))".to_string(),
mathematical_formulation: "\\text{erf}(x) = \\frac{2}{\\sqrt{\\pi}} \\sum_{n=0}^{\\infty} \\frac{(-1)^n x^{2n+1}}{n!(2n+1)}".to_string(),
conditions: vec![
"x ∈ ℂ".to_string(),
"Series converges for all finite x".to_string(),
"Radius of convergence is infinite".to_string(),
],
significance: "Enables accurate computation of error function values and provides insight into its analytical structure".to_string(),
historical_context: "This series was developed as part of the study of the normal distribution in probability theory".to_string(),
},
proof_steps: create_error_function_proof_steps(),
applications: vec![
Application {
title: "Probability Calculations".to_string(),
description: "Computing probabilities for normal distributions".to_string(),
mathematical_context: "P(X ≤ x) = (1/2)[1 + erf(x/√2)]".to_string(),
practical_example: "Computing P(|X| ≤ 1) for standard normal X".to_string(),
},
],
related_proofs: vec!["gaussian_integral_derivation".to_string()],
computational_examples: create_error_function_examples(),
});
}
fn run(&mut self) -> Result<(), Box<dyn std::error::Error>> {
println!("🧮 Interactive Proof Explorer for Special Functions");
println!("=====================================================\n");
println!("Welcome to the advanced mathematical proof exploration system!");
println!("Here you can learn and verify complex mathematical proofs step by step.\n");
self.display_main_menu()?;
loop {
println!("\n🔍 Main Menu:");
println!("1. 📚 Browse available proofs");
println!("2. 🎯 Continue a proof in progress");
println!("3. 📊 View your progress and achievements");
println!("4. ⚙️ Adjust difficulty settings");
println!("5. 🔍 Search proofs by topic");
println!("6. 💡 Get proof recommendations");
println!("7. 📖 Mathematical reference library");
println!("8. 💾 Save progress and exit");
let choice = get_user_input("Enter your choice (1-8): ")?;
match choice.parse::<u32>() {
Ok(1) => self.browse_proofs()?,
Ok(2) => self.continue_proof()?,
Ok(3) => self.display_progress()?,
Ok(4) => self.adjust_difficulty()?,
Ok(5) => self.search_proofs()?,
Ok(6) => self.get_recommendations()?,
Ok(7) => self.reference_library()?,
Ok(8) => {
self.save_progress()?;
println!("👋 Thank you for exploring mathematical proofs!");
break;
}
_ => println!("❌ Invalid choice. Please try again."),
}
}
Ok(())
}
fn display_main_menu(&self) -> Result<(), Box<dyn std::error::Error>> {
println!("📈 Your Current Status:");
println!(
"• Mathematical Level: {:?}",
self.user_progress.difficulty_level
);
println!(
"• Completed Proofs: {}",
self.user_progress.completed_proofs.len()
);
println!("• Available Proofs: {}", self.available_proofs.len());
if let Some(current) = &self.current_proof {
let proof = self
.get_proof_by_id(¤t.proof_id)
.expect("Operation failed");
println!(
"• Current Proof: {} (Step {}/{})",
proof.title,
current.current_step + 1,
proof.proof_steps.len()
);
}
Ok(())
}
fn browse_proofs(&self) -> Result<(), Box<dyn std::error::Error>> {
println!("\n📚 Available Mathematical Proofs");
println!("==================================\n");
for (i, proof) in self.available_proofs.iter().enumerate() {
let completed = if self.user_progress.completed_proofs.contains(&proof.id) {
"✅"
} else {
"🔒"
};
let level_indicator = match proof.mathematical_level {
MathematicalLevel::Undergraduate => "🟢",
MathematicalLevel::Graduate => "🟡",
MathematicalLevel::Advanced => "🟠",
MathematicalLevel::Research => "🔴",
};
println!(
"{}. {} {} {}",
i + 1,
completed,
level_indicator,
proof.title
);
println!(" Level: {:?}", proof.mathematical_level);
println!(" {}", proof.description);
println!(" Prerequisites: {}", proof.prerequisites.join(", "));
println!();
}
println!("Legend: ✅ Completed, 🔒 Available, 🟢 Undergraduate, 🟡 Graduate, 🟠 Advanced, 🔴 Research\n");
let choice = get_user_input("Enter proof number to start, or 'back' to return: ")?;
if choice.to_lowercase() == "back" {
return Ok(());
}
if let Ok(index) = choice.parse::<usize>() {
if index > 0 && index <= self.available_proofs.len() {
let proof = &self.available_proofs[index - 1];
self.start_proof_exploration(&proof.id)?;
} else {
println!("❌ Invalid proof number.");
}
}
Ok(())
}
fn start_proof_exploration(&self, proofid: &str) -> Result<(), Box<dyn std::error::Error>> {
let proof = self.get_proof_by_id(proofid).expect("Operation failed");
println!("\n🎯 Starting Proof: {}", proof.title);
println!("=================={}", "=".repeat(proof.title.len()));
println!();
println!("📜 Theorem Statement:");
println!("{}", proof.main_theorem.statement);
println!();
println!("🧮 Mathematical Formulation:");
println!("{}", proof.main_theorem.mathematical_formulation);
println!();
println!("📋 Conditions:");
for condition in &proof.main_theorem.conditions {
println!(" • {}", condition);
}
println!();
println!("🌟 Significance:");
println!("{}", proof.main_theorem.significance);
println!();
println!("📚 Historical Context:");
println!("{}", proof.main_theorem.historical_context);
println!();
let start = get_user_input("Ready to begin the proof? (y/n): ")?;
if start.to_lowercase() == "y" {
self.run_proof_session(proof)?;
}
Ok(())
}
fn run_proof_session(&self, proof: &ProofModule) -> Result<(), Box<dyn std::error::Error>> {
println!("\n🚀 Beginning Proof Session");
println!("===========================\n");
let session_start = Instant::now();
let mut current_step = 0;
let mut step_times = Vec::new();
let mut user_responses = Vec::new();
while current_step < proof.proof_steps.len() {
let step = &proof.proof_steps[current_step];
let step_start = Instant::now();
println!(
"📝 Step {} of {}: {}",
current_step + 1,
proof.proof_steps.len(),
step.title
);
println!("{}", "=".repeat(50));
println!();
println!("💡 Explanation:");
println!("{}", step.explanation);
println!();
println!("🧮 Mathematical Content:");
println!("{}", step.mathematical_content);
println!();
println!("📖 Justification:");
println!("{}", step.justification);
println!();
for (i, element) in step.interactive_elements.iter().enumerate() {
match self.handle_interactive_element(element, current_step, i)? {
ElementResult::Correct(response) => {
user_responses.push(UserResponse {
step_id: step.id.clone(),
response,
correct: true,
attempts: 1,
time_taken: step_start.elapsed(),
});
}
ElementResult::Incorrect(response, attempts) => {
user_responses.push(UserResponse {
step_id: step.id.clone(),
response,
correct: false,
attempts,
time_taken: step_start.elapsed(),
});
}
ElementResult::Skipped => {}
}
}
if self.difficulty_settings.require_step_verification {
if let Some(verification) = &step.verification_method {
self.run_verification(verification)?;
}
}
step_times.push(step_start.elapsed());
let continue_choice =
get_user_input("\nPress Enter to continue to next step, or 'quit' to exit: ")?;
if continue_choice.to_lowercase() == "quit" {
break;
}
current_step += 1;
println!("\n{}\n", "=".repeat(80));
}
if current_step == proof.proof_steps.len() {
self.celebrate_completion(proof, session_start.elapsed())?;
}
Ok(())
}
fn handle_interactive_element(
&self,
element: &InteractiveElement,
step: usize,
element_index: usize,
) -> Result<ElementResult, Box<dyn std::error::Error>> {
match element {
InteractiveElement::UserInput {
prompt,
expected_answer,
validation_tolerance,
hints,
} => {
println!("❓ {}", prompt);
let mut attempts = 0;
loop {
attempts += 1;
let response = get_user_input("Your answer: ")?;
if self.validate_numerical_response(
&response,
expected_answer,
*validation_tolerance,
) {
println!("✅ Correct!");
return Ok(ElementResult::Correct(response));
} else {
println!("❌ Not quite right.");
if attempts <= hints.len() && !hints.is_empty() {
println!("💡 Hint: {}", hints[attempts - 1]);
}
if attempts >= 3 {
println!("🔍 Expected answer: {}", expected_answer);
return Ok(ElementResult::Incorrect(response, attempts as u32));
}
}
}
}
InteractiveElement::Visualization {
title,
description,
plot_type,
} => {
println!("📊 Visualization: {}", title);
println!("{}", description);
self.create_visualization(plot_type)?;
Ok(ElementResult::Skipped)
}
InteractiveElement::Computation {
description,
code_example,
expected_result,
} => {
println!("💻 Computational Verification:");
println!("{}", description);
println!("Code example:");
println!("```rust");
println!("{}", code_example);
println!("```");
println!("Expected result: {}", expected_result);
let run_code = get_user_input("Run computation? (y/n): ")?;
if run_code.to_lowercase() == "y" {
self.execute_computational_example(code_example)?;
}
Ok(ElementResult::Skipped)
}
InteractiveElement::ConceptualQuestion {
question,
options,
correct_answer,
explanation,
} => {
println!("🤔 Conceptual Question:");
println!("{}", question);
println!();
for (i, option) in options.iter().enumerate() {
println!(" {}. {}", (b'A' + i as u8) as char, option);
}
let response = get_user_input("\nYour answer (A, B, C, etc.): ")?;
let answer_index = response.to_uppercase().chars().next().and_then(|c: char| {
if c.is_ascii_uppercase() {
Some((c as u8 - b'A') as usize)
} else {
None
}
});
if let Some(idx) = answer_index {
if idx == *correct_answer {
println!("✅ Correct!");
println!("💡 Explanation: {}", explanation);
return Ok(ElementResult::Correct(response));
} else {
println!("❌ Incorrect.");
println!("💡 Explanation: {}", explanation);
return Ok(ElementResult::Incorrect(response, 1));
}
}
Ok(ElementResult::Incorrect(response, 1))
}
}
}
fn validate_numerical_response(&self, response: &str, expected: &str, tolerance: f64) -> bool {
if let (Ok(user_val), Ok(expected_val)) = (response.parse::<f64>(), expected.parse::<f64>())
{
(user_val - expected_val).abs() <= tolerance
} else {
response.trim() == expected.trim()
}
}
fn create_visualization(&self, plottype: &PlotType) -> Result<(), Box<dyn std::error::Error>> {
match plottype {
PlotType::FunctionGraph { domain, functions } => {
println!("📈 Function Graph:");
println!("Domain: [{}, {}]", domain.0, domain.1);
println!("Functions: {}", functions.join(", "));
self.create_ascii_plot(domain, functions)?;
}
PlotType::ComplexPlane { radius, function } => {
println!("🌀 Complex Plane Visualization:");
println!("Function: {}", function);
println!("Radius: {}", radius);
self.create_complex_plane_plot(*radius, function)?;
}
PlotType::Convergence { series, terms } => {
println!("📊 Convergence Visualization:");
println!("Series: {}", series);
println!("Terms shown: {}", terms);
self.demonstrate_convergence(series, *terms)?;
}
PlotType::AsymptoticComparison {
exact,
approximation,
domain,
} => {
println!("📏 Asymptotic Comparison:");
println!("Exact: {}", exact);
println!("Approximation: {}", approximation);
println!("Domain: [{}, {}]", domain.0, domain.1);
self.compare_asymptotic_behavior(exact, approximation, domain)?;
}
}
wait_for_user_input()?;
Ok(())
}
fn create_ascii_plot(
&self,
domain: &(f64, f64),
functions: &[String],
) -> Result<(), Box<dyn std::error::Error>> {
println!("\nASCII Plot:");
let num_points = 40;
let step = (domain.1 - domain.0) / (num_points - 1) as f64;
println!("x f(x) Visualization");
println!("────────────────────────────────────");
for i in 0..num_points {
let x = domain.0 + i as f64 * step;
let y = if functions.contains(&"gamma".to_string()) {
gamma(x)
} else if functions.contains(&"sin".to_string()) {
x.sin()
} else {
x };
let normalized_y = ((y + 2.0) * 10.0) as usize;
let display_pos = normalized_y.min(20);
let mut line = [' '; 21];
line[10] = '|'; if display_pos < line.len() {
line[display_pos] = '●';
}
let display: String = line.iter().collect();
println!("{:5.1} {:8.3} {}", x, y, display);
}
Ok(())
}
fn create_complex_plane_plot(
&self,
radius: f64,
function: &str,
) -> Result<(), Box<dyn std::error::Error>> {
println!("\nComplex Plane (|z| ≤ {}):", radius);
for row in 0..15 {
for col in 0..30 {
let re = (col as f64 - 15.0) * radius / 15.0;
let im = (7.5 - row as f64) * radius / 7.5;
let z = Complex64::new(re, im);
if z.norm() <= radius {
let result = if function == "gamma" {
gamma_complex(z)
} else {
z };
let magnitude = result.norm();
let char = if magnitude < 1.0 {
'·'
} else if magnitude < 2.0 {
'▒'
} else if magnitude < 5.0 {
'▓'
} else {
'█'
};
print!("{}", char);
} else {
print!(" ");
}
}
println!();
}
Ok(())
}
fn demonstrate_convergence(
&self,
series: &str,
terms: usize,
) -> Result<(), Box<dyn std::error::Error>> {
println!("\nConvergence Demonstration for {}:", series);
println!("Term Partial Sum Error");
println!("──────────────────────────────");
if series.contains("gamma") {
let x: f64 = 10.0;
let exact = gamma(x);
let mut sum = (x - 0.5) * x.ln() - x + 0.5 * (2.0 * PI).ln();
println!(
"{:4} {:12.6} {:10.2e}",
0,
sum.exp(),
(sum.exp() - exact).abs()
);
for k in 1..=terms.min(5) {
let bernoulli_coeff = match k {
1 => 1.0 / 12.0,
2 => -1.0 / 360.0,
3 => 1.0 / 1260.0,
4 => -1.0 / 1680.0,
_ => 0.0,
};
sum += bernoulli_coeff / x.powi(2 * k as i32 - 1);
println!(
"{:4} {:12.6} {:10.2e}",
k,
sum.exp(),
(sum.exp() - exact).abs()
);
}
}
Ok(())
}
fn compare_asymptotic_behavior(
&self,
exact: &str,
approximation: &str,
domain: &(f64, f64),
) -> Result<(), Box<dyn std::error::Error>> {
println!("\nAsymptotic Comparison:");
println!("x Exact Approx Relative Error");
println!("────────────────────────────────────────────");
let num_points = 10;
let step = (domain.1 - domain.0) / (num_points - 1) as f64;
for i in 0..num_points {
let x = domain.0 + i as f64 * step;
let exact_val = if exact.contains("gamma") { gamma(x) } else { x };
let approx_val = if approximation.contains("stirling") {
((2.0 * PI / x).sqrt() * (x / std::f64::consts::E).powf(x)).exp()
} else {
exact_val
};
let rel_error = if exact_val != 0.0 {
((exact_val - approx_val) / exact_val).abs()
} else {
0.0
};
println!(
"{:8.1} {:10.4} {:10.4} {:12.2e}",
x, exact_val, approx_val, rel_error
);
}
Ok(())
}
fn execute_computational_example(&self, code: &str) -> Result<(), Box<dyn std::error::Error>> {
println!("\n💻 Executing computational example...");
if code.contains("gamma(") {
println!("Result: (executing gamma function computations)");
if code.contains("1/2") {
let result = gamma(0.5);
println!("gamma(1/2) = {:.10}", result);
println!("√π = {:.10}", PI.sqrt());
println!("Difference: {:.2e}", (result - PI.sqrt()).abs());
}
}
Ok(())
}
fn run_verification(
&self,
verification: &VerificationMethod,
) -> Result<(), Box<dyn std::error::Error>> {
println!("\n🔍 Verification Step:");
match verification {
VerificationMethod::NumericalCheck {
description,
test_cases,
} => {
println!("{}", description);
println!("\nRunning numerical tests:");
for (i, test_case) in test_cases.iter().enumerate() {
println!("Test {}: {}", i + 1, test_case.description);
let actual = gamma(test_case.input); let error = (actual - test_case.expected).abs();
let passed = error <= test_case.tolerance;
println!(" Input: {}", test_case.input);
println!(" Expected: {:.6}", test_case.expected);
println!(" Actual: {:.6}", actual);
println!(" Error: {:.2e}", error);
println!(" Result: {}", if passed { "✅ PASS" } else { "❌ FAIL" });
println!();
}
}
VerificationMethod::SymbolicVerification {
description,
identity,
} => {
println!("{}", description);
println!("Identity to verify: {}", identity);
println!("(Symbolic verification would be performed here)");
}
VerificationMethod::AsymptoticVerification {
description,
limit_expression,
} => {
println!("{}", description);
println!("Limit expression: {}", limit_expression);
println!("(Asymptotic verification would be performed here)");
}
}
wait_for_user_input()?;
Ok(())
}
fn celebrate_completion(
&self,
proof: &ProofModule,
total_time: Duration,
) -> Result<(), Box<dyn std::error::Error>> {
println!("\n🎉 Proof Completed Successfully!");
println!("===================================\n");
println!("🏆 Congratulations! You have successfully completed:");
println!("📜 {}", proof.title);
println!(
"⏱️ Total _time: {:.1} minutes",
total_time.as_secs_f64() / 60.0
);
println!("🎯 Mathematical level: {:?}", proof.mathematical_level);
println!();
println!("🌟 What you've learned:");
println!(" • Understanding of: {}", proof.main_theorem.statement);
println!(" • Complex analysis techniques");
println!(" • Connection between discrete and continuous mathematics");
println!();
println!("🔗 Related topics you might explore next:");
for related in &proof.related_proofs {
println!(" • {}", related);
}
println!();
println!("🏅 Achievement unlocked: Advanced Proof Explorer!");
wait_for_user_input()?;
Ok(())
}
fn continue_proof(&self) -> Result<(), Box<dyn std::error::Error>> {
if let Some(_current) = &self.current_proof {
println!("Continuing proof session...");
} else {
println!("No proof session in progress.");
}
Ok(())
}
fn display_progress(&self) -> Result<(), Box<dyn std::error::Error>> {
println!("\n📊 Your Learning Progress");
println!("==========================\n");
println!("🎯 Overall Statistics:");
println!(
"• Completed Proofs: {}",
self.user_progress.completed_proofs.len()
);
println!(
"• Mathematical Level: {:?}",
self.user_progress.difficulty_level
);
println!(
"• Total Study Time: {:.1} hours",
self.user_progress
.time_spent
.values()
.map(|d| d.as_secs_f64())
.sum::<f64>()
/ 3600.0
);
println!("\n🏆 Achievements:");
for achievement in &self.user_progress.achievements {
println!("• {}", achievement);
}
if self.user_progress.achievements.is_empty() {
println!("• Complete your first proof to earn achievements!");
}
wait_for_user_input()?;
Ok(())
}
fn adjust_difficulty(&mut self) -> Result<(), Box<dyn std::error::Error>> {
println!("\n⚙️ Difficulty Settings");
println!("=======================\n");
println!("Current settings:");
println!(
"• Mathematical rigor: {:?}",
self.difficulty_settings.mathematical_rigor_level
);
println!(
"• Show hints automatically: {}",
self.difficulty_settings.show_hints_automatically
);
println!(
"• Require step verification: {}",
self.difficulty_settings.require_step_verification
);
println!(
"• Include computational checks: {}",
self.difficulty_settings.include_computational_checks
);
wait_for_user_input()?;
Ok(())
}
fn search_proofs(&self) -> Result<(), Box<dyn std::error::Error>> {
println!("\n🔍 Search Proofs");
println!("=================\n");
let query = get_user_input("Enter search term (theorem name, topic, or keyword): ")?;
println!("Search results for '{}':", query);
for proof in &self.available_proofs {
if proof.title.to_lowercase().contains(&query.to_lowercase())
|| proof
.description
.to_lowercase()
.contains(&query.to_lowercase())
|| proof
.prerequisites
.iter()
.any(|p| p.to_lowercase().contains(&query.to_lowercase()))
{
println!("• {} ({:?})", proof.title, proof.mathematical_level);
println!(" {}", proof.description);
}
}
wait_for_user_input()?;
Ok(())
}
fn get_recommendations(&self) -> Result<(), Box<dyn std::error::Error>> {
println!("\n💡 Personalized Recommendations");
println!("=================================\n");
println!("Based on your current level and progress, we recommend:");
match self.user_progress.difficulty_level {
MathematicalLevel::Undergraduate => {
println!("• Start with the Error Function Series proof");
println!("• Build foundation with basic special function properties");
}
MathematicalLevel::Graduate => {
println!("• Explore the Gamma Function Reflection Formula");
println!("• Learn complex analysis techniques through Bessel functions");
}
_ => {
println!("• Challenge yourself with Stirling's Asymptotic Series");
println!("• Explore advanced topics in special function theory");
}
}
wait_for_user_input()?;
Ok(())
}
fn reference_library(&self) -> Result<(), Box<dyn std::error::Error>> {
println!("\n📖 Mathematical Reference Library");
println!("===================================\n");
println!("Available references:");
println!("1. 📚 Special Functions Handbook");
println!("2. 🧮 Complex Analysis Review");
println!("3. 📊 Asymptotic Methods");
println!("4. 🔢 Number Theory Connections");
println!("5. 🌐 Historical Development");
let choice = get_user_input("Select reference (1-5) or 'back': ")?;
match choice.as_str() {
"1" => self.show_special_functions_reference()?,
"2" => self.show_complex_analysis_reference()?,
"3" => self.show_asymptotic_methods_reference()?,
"4" => self.show_number_theory_reference()?,
"5" => self.show_historical_reference()?,
_ => return Ok(()),
}
Ok(())
}
fn show_special_functions_reference(&self) -> Result<(), Box<dyn std::error::Error>> {
println!("\n📚 Special Functions Handbook");
println!("==============================\n");
println!("🎲 Gamma Function:");
println!("• Definition: Γ(z) = ∫₀^∞ t^(z-1) e^(-t) dt");
println!("• Recurrence: Γ(z+1) = z·Γ(z)");
println!("• Reflection: Γ(z)Γ(1-z) = π/sin(πz)");
println!("• Special values: Γ(1/2) = √π, Γ(n) = (n-1)!");
println!();
println!("🌊 Bessel Functions:");
println!("• Differential equation: x²y'' + xy' + (x²-ν²)y = 0");
println!("• Generating function: exp(x(t-1/t)/2) = Σ J_n(x) t^n");
println!("• Orthogonality: ∫₀¹ J_n(αx)J_n(βx)x dx = 0 if α ≠ β");
println!();
println!("📊 Error Function:");
println!("• Definition: erf(x) = (2/√π) ∫₀^x e^(-t²) dt");
println!("• Complement: erfc(x) = 1 - erf(x)");
println!("• Series: erf(x) = (2/√π) Σ (-1)^n x^(2n+1) / (n!(2n+1))");
wait_for_user_input()?;
Ok(())
}
fn show_complex_analysis_reference(&self) -> Result<(), Box<dyn std::error::Error>> {
println!("\n🧮 Complex Analysis Review");
println!("============================\n");
println!("🔍 Key Concepts for Special Functions:");
println!("• Analytic continuation extends functions beyond original domains");
println!("• Residue calculus enables evaluation of complex integrals");
println!("• Branch cuts and Riemann surfaces handle multivalued functions");
println!("• Steepest descent method provides asymptotic expansions");
println!();
println!("🎯 Applications:");
println!("• Gamma function poles and residues");
println!("• Bessel function integral representations");
println!("• Hypergeometric function transformation formulas");
wait_for_user_input()?;
Ok(())
}
fn show_asymptotic_methods_reference(&self) -> Result<(), Box<dyn std::error::Error>> {
println!("\n📊 Asymptotic Methods");
println!("======================\n");
println!("🎯 Key Methods:");
println!("• Stirling's method for factorial-like functions");
println!("• Saddle point method for exponential integrals");
println!("• WKB approximation for differential equations");
println!("• Euler-Maclaurin formula for sum-integral relationships");
wait_for_user_input()?;
Ok(())
}
fn show_number_theory_reference(&self) -> Result<(), Box<dyn std::error::Error>> {
println!("\n🔢 Number Theory Connections");
println!("==============================\n");
println!("🌟 Special Functions in Number Theory:");
println!("• Riemann zeta function: ζ(s) = Σ 1/n^s");
println!("• Gamma function and factorials");
println!("• Bernoulli numbers in asymptotic series");
println!("• Euler's constant γ in logarithmic integrals");
wait_for_user_input()?;
Ok(())
}
fn show_historical_reference(&self) -> Result<(), Box<dyn std::error::Error>> {
println!("\n🌐 Historical Development");
println!("==========================\n");
println!("📅 Timeline:");
println!("• 1720s: Euler begins work on factorial extension");
println!("• 1812: Legendre introduces Γ notation");
println!("• 1824: Bessel develops his functions for astronomy");
println!("• 1856: Weierstrass proves infinite product formulas");
println!("• 1900s: Modern asymptotic theory develops");
wait_for_user_input()?;
Ok(())
}
fn save_progress(&self) -> Result<(), Box<dyn std::error::Error>> {
println!("💾 Saving progress...");
println!("✅ Progress saved successfully!");
Ok(())
}
fn get_proof_by_id(&self, id: &str) -> Option<&ProofModule> {
self.available_proofs.iter().find(|p| p.id == id)
}
}
#[allow(dead_code)]
#[derive(Debug)]
enum ElementResult {
Correct(String),
Incorrect(String, u32),
Skipped,
}
#[allow(dead_code)]
fn create_reflection_formula_proof_steps() -> Vec<ProofStep> {
vec![
ProofStep {
id: "step1".to_string(),
title: "Beta Function Connection".to_string(),
explanation: "We start by connecting the gamma function to the beta function, which will provide a pathway to the trigonometric identity.".to_string(),
mathematical_content: "B(z, 1-z) = ∫₀¹ t^(z-1)(1-t)^(-z) dt = Γ(z)Γ(1-z)/Γ(1) = Γ(z)Γ(1-z)".to_string(),
justification: "This follows from the integral representation of the beta function and the fact that Γ(1) = 1.".to_string(),
interactive_elements: vec![
InteractiveElement::UserInput {
prompt: "What is the value of Γ(1)?".to_string(),
expected_answer: "1".to_string(),
validation_tolerance: 0.01,
hints: vec!["Remember that Γ(n) = (n-1)! for positive integers".to_string()],
},
],
verification_method: Some(VerificationMethod::NumericalCheck {
description: "Verify the beta function identity numerically".to_string(),
test_cases: vec![
TestCase {
input: 0.5,
expected: PI,
tolerance: 1e-10,
description: "B(1/2, 1/2) should equal π".to_string(),
},
],
}),
common_confusions: vec![
"Don't confuse B(z,w) with Γ(z)Γ(w)".to_string(),
],
hints: vec![
"The beta function is symmetric: B(z,w) = B(w,z)".to_string(),
],
},
ProofStep {
id: "step2".to_string(),
title: "Integral Transformation".to_string(),
explanation: "We transform the beta integral using substitution to make it amenable to complex analysis techniques.".to_string(),
mathematical_content: "Using t = u/(1+u), we get B(z,1-z) = ∫₀^∞ u^(z-1)/(1+u) du".to_string(),
justification: "This substitution transforms the finite interval [0,1] to the infinite interval [0,∞), which is better suited for complex analysis.".to_string(),
interactive_elements: vec![
InteractiveElement::Computation {
description: "Verify the substitution t = u/(1+u) transforms the integral correctly".to_string(),
code_example: "// When t = u/(1+u), then dt = du/(1+u)²\n// and (1-t) = 1/(1+u)".to_string(),
expected_result: "The Jacobian gives the correct transformation".to_string(),
},
],
verification_method: None,
common_confusions: vec![
"Be careful with the Jacobian of the transformation".to_string(),
],
hints: vec![
"Work out dt in terms of du carefully".to_string(),
],
},
]
}
#[allow(dead_code)]
fn create_stirling_proof_steps() -> Vec<ProofStep> {
vec![
ProofStep {
id: "stirling1".to_string(),
title: "Euler-Maclaurin Setup".to_string(),
explanation: "We begin with the Euler-Maclaurin formula to relate sums to integrals, which is key to understanding the asymptotic behavior of the gamma function.".to_string(),
mathematical_content: "∑_{k=1}^n f(k) = ∫₁ⁿ f(x)dx + ½[f(1)+f(n)] + ∑_{k=1}^p B₂ₖ/(2k)! [f^(2k-1)(n)-f^(2k-1)(1)] + Rp".to_string(),
justification: "The Euler-Maclaurin formula provides the connection between discrete sums and continuous integrals through Bernoulli numbers.".to_string(),
interactive_elements: vec![
InteractiveElement::ConceptualQuestion {
question: "What is the key insight behind the Euler-Maclaurin formula?".to_string(),
options: vec![
"It approximates integrals with sums".to_string(),
"It relates sums to integrals plus correction terms".to_string(),
"It computes derivatives numerically".to_string(),
"It solves differential equations".to_string(),
],
correct_answer: 1,
explanation: "The formula expresses sums as integrals plus correction terms involving Bernoulli numbers".to_string(),
},
],
verification_method: None,
common_confusions: vec![
"The Bernoulli numbers appear with specific powers".to_string(),
],
hints: vec![
"Think about approximating discrete with continuous".to_string(),
],
},
]
}
#[allow(dead_code)]
fn create_bessel_generating_proof_steps() -> Vec<ProofStep> {
vec![
ProofStep {
id: "bessel1".to_string(),
title: "Exponential Series Expansion".to_string(),
explanation: "We expand the exponential function exp(x(t-1/t)/2) in powers of t to identify the coefficients as Bessel functions.".to_string(),
mathematical_content: "exp(x(t-1/t)/2) = exp(xt/2)exp(-x/(2t)) = ∑_{m=0}^∞ (xt/2)^m/m! ∑_{n=0}^∞ (-x/(2t))^n/n!".to_string(),
justification: "We use the product of two exponential series and collect terms by powers of t.".to_string(),
interactive_elements: vec![
InteractiveElement::Visualization {
title: "Bessel Function Generating Function".to_string(),
description: "Visualize how the generating function creates oscillating Bessel functions".to_string(),
plot_type: PlotType::FunctionGraph {
domain: (0.0, 20.0),
functions: vec!["J_0(x)".to_string(), "J_1(x)".to_string()],
},
},
],
verification_method: Some(VerificationMethod::NumericalCheck {
description: "Verify the generating function for small values".to_string(),
test_cases: vec![
TestCase {
input: 1.0,
expected: 0.765198,
tolerance: 1e-5,
description: "J_0(1) from generating function".to_string(),
},
],
}),
common_confusions: vec![
"Keep track of positive and negative powers of t".to_string(),
],
hints: vec![
"Use the Cauchy product formula for the series".to_string(),
],
},
]
}
#[allow(dead_code)]
fn create_error_function_proof_steps() -> Vec<ProofStep> {
vec![
ProofStep {
id: "erf1".to_string(),
title: "Series Expansion of Exponential".to_string(),
explanation: "We start with the series expansion of e^(-t²) and integrate term by term to obtain the error function series.".to_string(),
mathematical_content: "e^(-t²) = ∑_{n=0}^∞ (-1)^n t^(2n)/n!".to_string(),
justification: "This is the standard power series expansion of the exponential function with argument -t².".to_string(),
interactive_elements: vec![
InteractiveElement::UserInput {
prompt: "What is the coefficient of t⁴ in the expansion of e^(-t²)?".to_string(),
expected_answer: "1/2".to_string(),
validation_tolerance: 0.01,
hints: vec!["Use the formula (-1)^n/n! with n=2".to_string()],
},
InteractiveElement::Visualization {
title: "Error Function Convergence".to_string(),
description: "Watch how the series converges to the error function".to_string(),
plot_type: PlotType::Convergence {
series: "erf(x) series".to_string(),
terms: 10,
},
},
],
verification_method: Some(VerificationMethod::NumericalCheck {
description: "Verify series convergence for erf(1)".to_string(),
test_cases: vec![
TestCase {
input: 1.0,
expected: 0.842701,
tolerance: 1e-5,
description: "erf(1) from series".to_string(),
},
],
}),
common_confusions: vec![
"Don't forget the 2/√π factor in the error function definition".to_string(),
],
hints: vec![
"Integrate each term of the series separately".to_string(),
],
},
]
}
#[allow(dead_code)]
fn create_reflection_formula_examples() -> Vec<ComputationalExample> {
vec![
ComputationalExample {
title: "Verifying the Reflection Formula".to_string(),
description: "Numerical verification of Γ(z)Γ(1-z) = π/sin(πz)".to_string(),
code: "let z = 0.3;\nlet left_side = gamma(z) * gamma(1.0 - z);\nlet right_side = PI / (PI * z).sin();\nprintln!(\"Left: {}, Right: {}\", left_side, right_side);".to_string(),
expected_output: "Both sides should be approximately equal".to_string(),
learning_objective: "Understand how the reflection formula connects gamma and trigonometric functions".to_string(),
},
]
}
#[allow(dead_code)]
fn create_stirling_examples() -> Vec<ComputationalExample> {
vec![
ComputationalExample {
title: "Stirling Approximation Accuracy".to_string(),
description: "Compare Stirling approximation with exact gamma values".to_string(),
code: "for n in 5..20 {\n let exact = gamma(n as f64);\n let stirling = stirling_approx(n as f64);\n println!(\"n={}: exact={:.2e}, stirling={:.2e}\", n, exact, stirling);\n}".to_string(),
expected_output: "Stirling approximation becomes increasingly accurate for larger n".to_string(),
learning_objective: "See how asymptotic approximations work in practice".to_string(),
},
]
}
#[allow(dead_code)]
fn create_bessel_generating_examples() -> Vec<ComputationalExample> {
vec![
ComputationalExample {
title: "Generating Function Verification".to_string(),
description: "Verify that the generating function produces correct Bessel function values".to_string(),
code: "let x = 2.0;\nlet t = 1.0;\nlet gen_func = (x * (t - 1.0/t) / 2.0).exp();\nlet series_sum = j0(x) + j1(x) * t + jminus1(x) / t;\nprintln!(\"Gen func: {}, Series: {}\", gen_func, series_sum);".to_string(),
expected_output: "The generating function equals the series sum".to_string(),
learning_objective: "Understand how generating functions encode infinite families of functions".to_string(),
},
]
}
#[allow(dead_code)]
fn create_error_function_examples() -> Vec<ComputationalExample> {
vec![
ComputationalExample {
title: "Error Function Series Convergence".to_string(),
description: "Demonstrate convergence of the error function power series".to_string(),
code: "let x = 1.0;\nlet mut sum = 0.0;\nfor n in 0..20 {\n let term = (-1.0_f64).powi(n) * x.powi(2*n+1) / (factorial(n) * (2*n+1));\n sum += term;\n println!(\"n={}: term={:.6e}, sum={:.6}\", n, term, 2.0/PI.sqrt() * sum);\n}".to_string(),
expected_output: "Series converges to erf(1) ≈ 0.8427".to_string(),
learning_objective: "See how infinite series represent transcendental functions".to_string(),
},
]
}
#[allow(dead_code)]
fn wait_for_user_input() -> Result<(), Box<dyn std::error::Error>> {
get_user_input("Press Enter to continue...")?;
Ok(())
}
#[allow(dead_code)]
fn get_user_input(prompt: &str) -> Result<String, Box<dyn std::error::Error>> {
print!("{}", prompt);
io::stdout().flush()?;
let mut input = String::new();
io::stdin().read_line(&mut input)?;
Ok(input.trim().to_string())
}
#[allow(dead_code)]
fn gamma_complex(z: Complex64) -> Complex64 {
if z.re > 0.0 {
let g = 7.0;
let coef = [
0.999_999_999_999_809_9,
676.5203681218851,
-1259.1392167224028,
771.323_428_777_653_1,
-176.615_029_162_140_6,
12.507343278686905,
-0.13857109526572012,
9.984_369_578_019_572e-6,
1.5056327351493116e-7,
];
let z_shifted = z - Complex64::new(1.0, 0.0);
let mut x = Complex64::new(coef[0], 0.0);
for i in 1..coef.len() {
x += Complex64::new(coef[i], 0.0) / (z_shifted + Complex64::new(i as f64, 0.0));
}
let t = z_shifted + Complex64::new(g + 0.5, 0.0);
let sqrt_2pi = Complex64::new((2.0 * PI).sqrt(), 0.0);
sqrt_2pi * t.powc(z_shifted + Complex64::new(0.5, 0.0)) * (-t).exp() * x
} else {
let pi_z = Complex64::new(PI, 0.0) * z;
let sin_pi_z = pi_z.sin();
if sin_pi_z.norm() < 1e-15 {
Complex64::new(f64::INFINITY, 0.0)
} else {
Complex64::new(PI, 0.0) / (sin_pi_z * gamma_complex(Complex64::new(1.0, 0.0) - z))
}
}
}
#[allow(dead_code)]
fn main() -> Result<(), Box<dyn std::error::Error>> {
let mut explorer = ProofExplorer::new();
explorer.run()
}