//! # Topos Theory
//!
//! A topos is a category with additional structure that makes it behave
//! like a generalized universe of sets. It provides an internal logic
//! for reasoning about mathematical structures.
//!
//! ## Key Features
//!
//! - **Subobject classifier**: An object Ω with a universal property
//!   for classifying subobjects (generalizes {true, false} in Set)
//! - **Internal logic**: Intuitionistic logic derived from the topos structure
//! - **Exponentials**: All function spaces exist
//! - **Limits and colimits**: All finite limits and colimits exist

use crate::category::{
    Category, CategoryWithMono, CategoryWithProducts, CartesianClosedCategory,
    Object, ObjectData, Morphism, MorphismData,
};
use crate::{CategoryError, MorphismId, ObjectId, Result};
use dashmap::DashMap;
use serde::{Deserialize, Serialize};
use std::collections::HashMap;
use std::sync::Arc;

/// The subobject classifier Ω
///
/// In a topos, the subobject classifier has a characteristic morphism
/// true: 1 -> Ω such that for any monomorphism m: A >-> B, there exists
/// a unique χ_m: B -> Ω making the pullback square commute.
#[derive(Debug, Clone, Serialize, Deserialize)]
pub struct SubobjectClassifier<T: Clone + std::fmt::Debug> {
    /// The classifier object Ω
    pub omega: Object<T>,
    /// The terminal object 1
    pub terminal: Object<T>,
    /// The truth morphism: true: 1 -> Ω
    pub truth: MorphismId,
    /// Cached characteristic morphisms
    pub characteristics: HashMap<MorphismId, MorphismId>,
}

impl<T: Clone + std::fmt::Debug + PartialEq> SubobjectClassifier<T> {
    /// Creates a new subobject classifier
    pub fn new(omega: Object<T>, terminal: Object<T>, truth: MorphismId) -> Self {
        Self {
            omega,
            terminal,
            truth,
            characteristics: HashMap::new(),
        }
    }

    /// Registers a characteristic morphism for a monomorphism
    pub fn register_characteristic(&mut self, mono: MorphismId, chi: MorphismId) {
        self.characteristics.insert(mono, chi);
    }

    /// Gets the characteristic morphism for a monomorphism
    pub fn characteristic_of(&self, mono: &MorphismId) -> Option<MorphismId> {
        self.characteristics.get(mono).copied()
    }
}

/// A topos is a category with special structure
///
/// Key properties:
/// 1. Has all finite limits
/// 2. Has all finite colimits
/// 3. Is cartesian closed (has exponentials)
/// 4. Has a subobject classifier
#[derive(Debug)]
pub struct Topos<C: Category> {
    /// The underlying category
    pub category: C,
    /// The subobject classifier
    subobject_classifier: Option<SubobjectClassifier<ObjectData>>,
    /// Truth values in the internal logic
    truth_values: Vec<MorphismId>,
    /// Cached exponential objects
    exponentials: Arc<DashMap<(ObjectId, ObjectId), ObjectId>>,
    /// Cached pullbacks
    pullbacks: Arc<DashMap<(MorphismId, MorphismId), PullbackData>>,
}

/// Data for a pullback square
#[derive(Debug, Clone, Serialize, Deserialize)]
pub struct PullbackData {
    /// The pullback object P
    pub pullback: ObjectId,
    /// First projection P -> A
    pub proj1: MorphismId,
    /// Second projection P -> B
    pub proj2: MorphismId,
}

impl<C: Category> Topos<C> {
    /// Creates a new topos from a category
    ///
    /// Note: This does not verify that the category actually forms a topos.
    /// Use `verify_topos_axioms` to check.
    pub fn new(category: C) -> Self {
        Self {
            category,
            subobject_classifier: None,
            truth_values: Vec::new(),
            exponentials: Arc::new(DashMap::new()),
            pullbacks: Arc::new(DashMap::new()),
        }
    }

    /// Sets the subobject classifier
    pub fn with_subobject_classifier(
        mut self,
        classifier: SubobjectClassifier<ObjectData>,
    ) -> Self {
        self.subobject_classifier = Some(classifier);
        self
    }

    /// Gets the subobject classifier if it exists
    pub fn subobject_classifier(&self) -> Option<&SubobjectClassifier<ObjectData>> {
        self.subobject_classifier.as_ref()
    }

    /// Adds a truth value
    pub fn add_truth_value(&mut self, morphism: MorphismId) {
        self.truth_values.push(morphism);
    }

    /// Gets all truth values
    pub fn truth_values(&self) -> &[MorphismId] {
        &self.truth_values
    }

    /// Gets the underlying category
    pub fn category(&self) -> &C {
        &self.category
    }
}

impl<C: Category + CategoryWithProducts> Topos<C> {
    /// Computes a pullback of f: A -> C and g: B -> C
    ///
    /// Returns the pullback object P with projections
    /// such that the square commutes.
    pub fn pullback(
        &self,
        f: &C::Morphism,
        g: &C::Morphism,
    ) -> Option<(C::Object, C::Morphism, C::Morphism)> {
        // Check that f and g have the same codomain
        if self.category.codomain(f) != self.category.codomain(g) {
            return None;
        }

        // For a concrete implementation, we would compute the actual pullback
        // This is a simplified version using products as an approximation
        let a = self.category.domain(f);
        let b = self.category.domain(g);

        // P is a subobject of A x B
        let product = self.category.product(&a, &b)?;
        let p1 = self.category.proj1(&product)?;
        let p2 = self.category.proj2(&product)?;

        Some((product, p1, p2))
    }

    /// Computes the equalizer of f, g: A -> B
    ///
    /// The equalizer E is the largest subobject of A where f = g
    pub fn equalizer(
        &self,
        f: &C::Morphism,
        g: &C::Morphism,
    ) -> Option<(C::Object, C::Morphism)> {
        // f and g must have the same domain and codomain
        if self.category.domain(f) != self.category.domain(g) {
            return None;
        }
        if self.category.codomain(f) != self.category.codomain(g) {
            return None;
        }

        // Simplified: return domain with identity if f = g
        // A real implementation would compute the actual equalizer
        let a = self.category.domain(f);
        let id = self.category.identity(&a)?;

        Some((a, id))
    }
}

impl<C: Category + CategoryWithProducts + CategoryWithMono> Topos<C> {
    /// Verifies that this is a valid topos
    ///
    /// Checks:
    /// 1. Finite limits exist (simplified: products and equalizers)
    /// 2. Has subobject classifier
    /// 3. Is cartesian closed (simplified check)
    pub fn verify_topos_axioms(&self) -> ToposVerification {
        let mut verification = ToposVerification::new();

        // Check subobject classifier
        if self.subobject_classifier.is_some() {
            verification.has_subobject_classifier = true;
        }

        // Check products (simplified)
        let objects = self.category.objects();
        if objects.len() >= 2 {
            let a = &objects[0];
            let b = &objects[1];
            verification.has_finite_products = self.category.product(a, b).is_some();
        }

        // Check for terminal object (simplified)
        verification.has_terminal = !objects.is_empty();

        verification
    }
}

/// Result of topos axiom verification
#[derive(Debug, Clone, Serialize, Deserialize)]
pub struct ToposVerification {
    pub has_subobject_classifier: bool,
    pub has_finite_products: bool,
    pub has_finite_coproducts: bool,
    pub has_equalizers: bool,
    pub has_coequalizers: bool,
    pub has_terminal: bool,
    pub has_initial: bool,
    pub is_cartesian_closed: bool,
}

impl ToposVerification {
    pub fn new() -> Self {
        Self {
            has_subobject_classifier: false,
            has_finite_products: false,
            has_finite_coproducts: false,
            has_equalizers: false,
            has_coequalizers: false,
            has_terminal: false,
            has_initial: false,
            is_cartesian_closed: false,
        }
    }

    pub fn is_topos(&self) -> bool {
        self.has_subobject_classifier
            && self.has_finite_products
            && self.has_terminal
            && self.is_cartesian_closed
    }
}

impl Default for ToposVerification {
    fn default() -> Self {
        Self::new()
    }
}

/// Internal logic operations in a topos
///
/// The subobject classifier Ω supports logical operations
/// that form an internal Heyting algebra.
#[derive(Debug)]
pub struct InternalLogic {
    /// Conjunction: ∧: Ω x Ω -> Ω
    pub conjunction: Option<MorphismId>,
    /// Disjunction: ∨: Ω x Ω -> Ω
    pub disjunction: Option<MorphismId>,
    /// Implication: →: Ω x Ω -> Ω
    pub implication: Option<MorphismId>,
    /// Negation: ¬: Ω -> Ω
    pub negation: Option<MorphismId>,
    /// Universal quantifier for each object
    pub universal: HashMap<ObjectId, MorphismId>,
    /// Existential quantifier for each object
    pub existential: HashMap<ObjectId, MorphismId>,
}

impl InternalLogic {
    pub fn new() -> Self {
        Self {
            conjunction: None,
            disjunction: None,
            implication: None,
            negation: None,
            universal: HashMap::new(),
            existential: HashMap::new(),
        }
    }

    /// Checks if the logic is complete (all operations defined)
    pub fn is_complete(&self) -> bool {
        self.conjunction.is_some()
            && self.disjunction.is_some()
            && self.implication.is_some()
            && self.negation.is_some()
    }

    /// Checks if the logic is classical (excluded middle holds)
    /// In general, topos logic is intuitionistic
    pub fn is_classical(&self) -> bool {
        // Would need to verify ¬¬p = p for all p
        // By default, topos logic is intuitionistic
        false
    }
}

impl Default for InternalLogic {
    fn default() -> Self {
        Self::new()
    }
}

/// A subobject in a topos
///
/// Subobjects are equivalence classes of monomorphisms into an object
#[derive(Debug, Clone, Serialize, Deserialize)]
pub struct Subobject {
    /// The source object of the monomorphism
    pub source: ObjectId,
    /// The target object (what we're a subobject of)
    pub target: ObjectId,
    /// The monomorphism
    pub mono: MorphismId,
    /// The characteristic morphism χ: target -> Ω
    pub characteristic: Option<MorphismId>,
}

impl Subobject {
    pub fn new(source: ObjectId, target: ObjectId, mono: MorphismId) -> Self {
        Self {
            source,
            target,
            mono,
            characteristic: None,
        }
    }

    pub fn with_characteristic(mut self, chi: MorphismId) -> Self {
        self.characteristic = Some(chi);
        self
    }
}

/// Lattice of subobjects for an object in a topos
///
/// In a topos, the subobjects of any object form a Heyting algebra
#[derive(Debug)]
pub struct SubobjectLattice {
    /// The object whose subobjects we're tracking
    pub object: ObjectId,
    /// All subobjects (ordered by inclusion)
    pub subobjects: Vec<Subobject>,
    /// Meet (intersection) results
    meets: HashMap<(usize, usize), usize>,
    /// Join (union) results
    joins: HashMap<(usize, usize), usize>,
}

impl SubobjectLattice {
    pub fn new(object: ObjectId) -> Self {
        Self {
            object,
            subobjects: Vec::new(),
            meets: HashMap::new(),
            joins: HashMap::new(),
        }
    }

    /// Adds a subobject to the lattice
    pub fn add(&mut self, subobject: Subobject) -> usize {
        let index = self.subobjects.len();
        self.subobjects.push(subobject);
        index
    }

    /// Computes the meet (intersection) of two subobjects
    pub fn meet(&self, a: usize, b: usize) -> Option<usize> {
        self.meets.get(&(a.min(b), a.max(b))).copied()
    }

    /// Computes the join (union) of two subobjects
    pub fn join(&self, a: usize, b: usize) -> Option<usize> {
        self.joins.get(&(a.min(b), a.max(b))).copied()
    }

    /// Records a meet computation
    pub fn record_meet(&mut self, a: usize, b: usize, result: usize) {
        self.meets.insert((a.min(b), a.max(b)), result);
    }

    /// Records a join computation
    pub fn record_join(&mut self, a: usize, b: usize, result: usize) {
        self.joins.insert((a.min(b), a.max(b)), result);
    }
}

#[cfg(test)]
mod tests {
    use super::*;
    use crate::category::SetCategory;

    #[test]
    fn test_topos_creation() {
        let cat = SetCategory::new();
        let topos = Topos::new(cat);

        assert!(topos.subobject_classifier().is_none());
    }

    #[test]
    fn test_subobject_classifier() {
        let omega = Object::new(ObjectData::FiniteSet(2)); // {false, true}
        let terminal = Object::new(ObjectData::Terminal);
        let truth = MorphismId::new();

        let classifier = SubobjectClassifier::new(omega, terminal, truth);

        assert_eq!(classifier.omega.data, ObjectData::FiniteSet(2));
    }

    #[test]
    fn test_internal_logic() {
        let logic = InternalLogic::new();

        assert!(!logic.is_complete());
        assert!(!logic.is_classical());
    }

    #[test]
    fn test_subobject() {
        let source = ObjectId::new();
        let target = ObjectId::new();
        let mono = MorphismId::new();

        let sub = Subobject::new(source, target, mono);

        assert_eq!(sub.source, source);
        assert!(sub.characteristic.is_none());
    }

    #[test]
    fn test_topos_verification() {
        let verification = ToposVerification::new();

        assert!(!verification.is_topos());
    }
}