// ============================================================================
// CATEGORICAL HOUDINI: VEX Mathematical Library
// A comprehensive VEX implementation for category theory in 3D space
// ============================================================================

#pragma once

// ============================================================================
// CORE CATEGORICAL STRUCTURES
// ============================================================================

// Category object representation
struct CategoryObject {
    int id;                    // Unique identifier
    vector position;           // Spatial embedding
    string type;              // Object type (for typed categories)
    float@properties[];       // Extensible property array
}

// Morphism representation  
struct Morphism {
    int source_id;            // Domain object
    int target_id;            // Codomain object
    vector direction;         // Spatial direction vector
    string composition_rule;   // How this morphism composes
    float strength;           // Morphism "intensity"
}

// Functor implementation
struct Functor {
    string name;              // Functor identifier
    string source_category;   // Domain category
    string target_category;   // Codomain category
    float preservation_law;   // How well it preserves structure
}

// ============================================================================
// KAN EXTENSIONS: The Heart of Categorical Computation
// ============================================================================

/**
 * Right Kan Extension Implementation
 * Lan_K F: Extends functor F along K via colimit construction
 * 
 * Mathematical Definition:
 * (Lan_K F)(e) = colim_{K(c)→e} F(c)
 */
function float right_kan_extension(
    int point_id;           // Current point e in target category
    string functor_name;    // Functor F to extend
    float extension_radius; // Spatial radius for colimit computation
    int preserve_laws       // Whether to enforce categorical laws
) {
    vector current_pos = point(0, "P", point_id);
    float colimit_value = 0.0;
    float total_weight = 0.0;
    int violation_count = 0;
    
    // Step 1: Collect all morphisms K(c) → e within radius
    for(int pt = 0; pt < npoints(0); pt++) {
        vector source_pos = point(0, "P", pt);
        float distance_to_source = distance(current_pos, source_pos);
        
        if(distance_to_source <= extension_radius) {
            // Get original functor value F(c)
            float original_value = point(0, "functor_value", pt);
            
            // Compute weight based on categorical proximity
            float categorical_weight = 1.0 / max(distance_to_source, 0.001);
            
            // Check if this preserves composition laws
            if(preserve_laws) {
                float composition_error = check_composition_preservation(pt, point_id);
                if(composition_error > 0.1) {
                    violation_count++;
                    categorical_weight *= 0.1; // Penalize law violations
                }
            }
            
            colimit_value += original_value * categorical_weight;
            total_weight += categorical_weight;
        }
    }
    
    // Store violation information for visualization
    if(preserve_laws) {
        setpointattrib(0, "law_violations", point_id, violation_count);
    }
    
    return (total_weight > 0) ? colimit_value / total_weight : 0.0;
}

/**
 * Left Kan Extension Implementation  
 * Ran_K F: Extends functor F along K via limit construction
 */
function float left_kan_extension(
    int point_id;
    string functor_name;
    float extension_radius;
    int preserve_laws
) {
    vector current_pos = point(0, "P", point_id);
    float limit_value = 1e6; // Start with "infinity" for limit
    int found_sources = 0;
    
    // Step 1: Find all sources that map to current point via K
    for(int pt = 0; pt < npoints(0); pt++) {
        vector source_pos = point(0, "P", pt);
        float distance_to_source = distance(current_pos, source_pos);
        
        if(distance_to_source <= extension_radius) {
            float original_value = point(0, "functor_value", pt);
            
            // For limit, take minimum (categorical limit)
            limit_value = min(limit_value, original_value);
            found_sources = 1;
            
            // Composition law checking
            if(preserve_laws) {
                float composition_error = check_composition_preservation(pt, point_id);
                if(composition_error > 0.1) {
                    // Mark as problematic
                    setpointattrib(0, "limit_violation", point_id, 1);
                }
            }
        }
    }
    
    return found_sources ? limit_value : 0.0;
}

// ============================================================================
// TOPOLOGICAL INVARIANT COMPUTATION
// ============================================================================

/**
 * Fundamental Group π₁ Tracker
 * Computes homotopy classes of animation loops
 */
function int compute_fundamental_group_element(
    vector start_position;
    vector end_position;
    float time_parameter;
    string space_type
) {
    // For now, implement for common spaces
    if(space_type == "sphere_S2") {
        // S² has trivial fundamental group: π₁(S²) = 0
        return 0;
    }
    else if(space_type == "torus_T2") {
        // T² has π₁(T²) = Z × Z
        vector displacement = end_position - start_position;
        int winding_x = int(displacement.x / (2 * PI));
        int winding_y = int(displacement.y / (2 * PI));
        
        // Encode (m,n) ∈ Z×Z as single integer
        return winding_x * 1000 + winding_y;
    }
    else if(space_type == "klein_bottle") {
        // Klein bottle: π₁(K) = ⟨a,b | aba⁻¹b⟩
        // Simplified encoding for visualization
        vector normalized = normalize(end_position - start_position);
        return int(atan2(normalized.y, normalized.x) * 180 / PI);
    }
    
    return -1; // Unknown space type
}

/**
 * Homotopy Class Visualizer
 * Colors geometry based on homotopy equivalence
 */
function vector get_homotopy_color(int homotopy_class) {
    // Generate distinct colors for different homotopy classes
    float hue = (homotopy_class * 137.5) % 360; // Golden angle for distribution
    float saturation = 0.8;
    float value = 0.9;
    
    return hsv_to_rgb(set(hue, saturation, value));
}

// ============================================================================
// DEPENDENT TYPE SYSTEM FOR LOGICAL VOXELS
// ============================================================================

/**
 * Type Universe for Logical Propositions
 */
function string infer_logical_type(float truth_value; string proof_term) {
    if(proof_term == "") {
        return "proposition";  // Unproven proposition
    }
    else if(truth_value == 1.0) {
        return "theorem";      // Proven true statement
    }
    else if(truth_value == 0.0) {
        return "contradiction"; // Proven false
    }
    else {
        return "partial_proof"; // Incomplete proof
    }
}

/**
 * Type Checker for Logical Consistency
 */
function int type_check_logical_voxel(
    string logical_type;
    float truth_value;
    string proof_term;
    vector inference_direction
) {
    // Type consistency rules
    if(logical_type == "theorem" && truth_value != 1.0) {
        return 0; // Type error: theorems must be true
    }
    
    if(logical_type == "contradiction" && truth_value != 0.0) {
        return 0; // Type error: contradictions must be false
    }
    
    if(logical_type == "proposition" && proof_term != "") {
        return 0; // Type error: propositions shouldn't have proofs
    }
    
    // Check proof term validity (simplified)
    if(proof_term == "modus_ponens" || proof_term == "universal_instantiation" ||
       proof_term == "existential_generalization" || proof_term == "") {
        return 1; // Valid proof term
    }
    
    return 0; // Invalid proof term
}

/**
 * Inference Rule Application
 * Propagates logical truth through space
 */
function float apply_inference_rule(
    string rule_name;
    float premise1;
    float premise2;
    vector spatial_context
) {
    if(rule_name == "modus_ponens") {
        // If P and (P → Q), then Q
        return min(premise1, premise2);
    }
    else if(rule_name == "conjunction") {
        // P ∧ Q
        return min(premise1, premise2);
    }
    else if(rule_name == "disjunction") {
        // P ∨ Q  
        return max(premise1, premise2);
    }
    else if(rule_name == "spatial_propagation") {
        // Truth decays with spatial distance
        float decay_rate = 0.1;
        float distance_factor = length(spatial_context);
        return premise1 * exp(-decay_rate * distance_factor);
    }
    
    return 0.0; // Unknown rule
}

// ============================================================================
// COMPOSITION AND CATEGORICAL LAWS
// ============================================================================

/**
 * Check Composition Preservation
 * Verifies that Kan extensions preserve categorical structure
 */
function float check_composition_preservation(int source_id; int target_id) {
    // Get morphism chain from source to target
    vector source_pos = point(0, "P", source_id);
    vector target_pos = point(0, "P", target_id);
    
    // Check if there's an intermediate point for composition
    float min_composition_error = 1e6;
    
    for(int intermediate = 0; intermediate < npoints(0); intermediate++) {
        if(intermediate == source_id || intermediate == target_id) continue;
        
        vector intermediate_pos = point(0, "P", intermediate);
        
        // Check if intermediate lies on path from source to target
        float total_distance = distance(source_pos, target_pos);
        float path_distance = distance(source_pos, intermediate_pos) + 
                             distance(intermediate_pos, target_pos);
        
        if(abs(path_distance - total_distance) < 0.1) {
            // This could be a composition path
            float functor_source = point(0, "functor_value", source_id);
            float functor_intermediate = point(0, "functor_value", intermediate);
            float functor_target = point(0, "functor_value", target_id);
            
            // Check if F(g ∘ f) = F(g) ∘ F(f)
            float composition_direct = functor_target;
            float composition_chain = functor_intermediate * functor_source;
            
            float error = abs(composition_direct - composition_chain);
            min_composition_error = min(min_composition_error, error);
        }
    }
    
    return min_composition_error;
}

/**
 * Identity Morphism Checker
 * Ensures every object has an identity morphism
 */
function int has_identity_morphism(int object_id) {
    float self_functor = point(0, "functor_value", object_id);
    
    // Identity should preserve the object
    return (abs(self_functor - 1.0) < 0.001) ? 1 : 0;
}

// ============================================================================
// VISUALIZATION AND PROOF RENDERING
// ============================================================================

/**
 * Proof Surface Generator
 * Creates visual representations of logical proofs
 */
function vector generate_proof_surface_color(
    string logical_type;
    float truth_value;
    int type_correct;
    float proof_strength
) {
    vector base_color = {0.5, 0.5, 0.5}; // Default gray
    
    if(!type_correct) {
        return {1, 0, 0}; // Red for type errors
    }
    
    if(logical_type == "theorem") {
        base_color = {0, 1, 0}; // Green for proven theorems
    }
    else if(logical_type == "proposition") {
        base_color = {0, 0, 1}; // Blue for unproven propositions
    }
    else if(logical_type == "contradiction") {
        base_color = {1, 0, 1}; // Magenta for contradictions
    }
    else if(logical_type == "partial_proof") {
        base_color = {1, 1, 0}; // Yellow for partial proofs
    }
    
    // Modulate intensity by proof strength
    return base_color * proof_strength;
}

/**
 * Categorical Invariant Display
 * Shows mathematical properties as visual attributes
 */
function void display_categorical_invariants(int point_id) {
    // Store key invariants as point attributes for visualization
    float extension_quality = point(0, "extension_value", point_id) - 
                             point(0, "functor_value", point_id);
    
    setpointattrib(0, "extension_quality", point_id, extension_quality);
    setpointattrib(0, "categorical_rank", point_id, abs(extension_quality));
    
    // Homotopy information
    int homotopy_class = point(0, "homotopy_class", point_id);
    vector homotopy_color = get_homotopy_color(homotopy_class);
    setpointattrib(0, "homotopy_color", point_id, homotopy_color);
    
    // Type checking results
    string logical_type = point(0, "logical_type", point_id);
    float truth_value = point(0, "truth_value", point_id);
    string proof_term = point(0, "proof_term", point_id);
    
    int type_valid = type_check_logical_voxel(logical_type, truth_value, 
                                            proof_term, {0,0,0});
    setpointattrib(0, "type_valid", point_id, type_valid);
}

// ============================================================================
// MAIN EXECUTION FRAMEWORK
// ============================================================================

/**
 * Master Categorical Processor
 * Orchestrates all mathematical computations
 */
function void process_categorical_geometry() {
    // Get user parameters
    float kan_radius = chf("kan_extension_radius");
    int preserve_laws = chi("preserve_categorical_laws");
    string computation_mode = chs("computation_mode");
    
    // For each point, compute categorical extensions
    if(computation_mode == "right_kan_extension") {
        f@extension_value = right_kan_extension(@ptnum, "default_functor", 
                                              kan_radius, preserve_laws);
    }
    else if(computation_mode == "left_kan_extension") {
        f@extension_value = left_kan_extension(@ptnum, "default_functor", 
                                             kan_radius, preserve_laws);
    }
    
    // Compute topological invariants
    string space_type = chs("space_type");
    vector start_pos = point(0, "P", 0);
    i@homotopy_class = compute_fundamental_group_element(start_pos, @P, 
                                                        @Time, space_type);
    
    // Apply type checking for logical voxels
    s@logical_type = infer_logical_type(@truth_value, s@proof_term);
    i@type_valid = type_check_logical_voxel(s@logical_type, @truth_value, 
                                          s@proof_term, v@inference_direction);
    
    // Generate visualization
    display_categorical_invariants(@ptnum);
    
    // Set final color based on all computations
    v@Cd = generate_proof_surface_color(s@logical_type, @truth_value, 
                                       i@type_valid, @extension_value);
}

// Execute the main framework
process_categorical_geometry();