基于模态同伦类型论的定理证明器

Valkyrie 内置了基于模态同伦类型论（Modal Homotopy Type Theory）的定理证明器，提供了现代数学基础的形式化验证能力。通过类型即命题的对应关系，可以在类型系统中直接进行数学证明。

核心概念

同伦类型论基础

# 基础类型宇宙
Universe Type : Type₁
Universe Prop : Type₀

# 恒等类型（路径类型）
structure Path<A: Type, x: A, y: A> : Type {
    # 路径归纳原理
    refl(x: A) -> Path<A, x, x>
}

# 路径操作
micro path_concat<A: Type, x: A, y: A, z: A>(
    p: Path<A, x, y>, 
    q: Path<A, y, z>
) -> Path<A, x, z>

micro path_inverse<A: Type, x: A, y: A>(
    p: Path<A, x, y>
) -> Path<A, y, x>

# 函数外延性
axiom funext<A: Type, B: A -> Type, f: (x: A) -> B(x), g: (x: A) -> B(x)>(
    h: (x: A) -> Path<B(x), f(x), g(x)>
) -> Path<(x: A) -> B(x), f, g>

# 单价性公理
axiom univalence<A: Type, B: Type>(
    f: A ≃ B  # 等价关系
) -> Path<Type, A, B>

模态类型系统

# 模态算子
structure Modal<M: Modality, A: Type> : Type

# 模态系统基础
trait Modality {
    # 模态的基本属性
    axioms: ModalAxioms,
    # 模态转换规则
    transitions: ModalTransitions
}

# 内置模态实现
structure Necessity : Modality {
    axioms: S4Axioms,  # □p → p, □p → □□p
    transitions: NecessityRules
}

structure Possibility : Modality {
    axioms: S5Axioms,  # ◇p ↔ ¬□¬p
    transitions: PossibilityRules
}

structure Temporal<T: TimePoint> : Modality {
    axioms: TemporalAxioms<T>,
    transitions: TemporalRules<T>
}

structure Epistemic<A: Agent> : Modality {
    axioms: EpistemicAxioms<A>,
    transitions: KnowledgeRules<A>
}

# 用户可扩展的模态定义
structure CustomModal<Axioms: ModalAxioms, Rules: ModalTransitions> : Modality {
    axioms: Axioms,
    transitions: Rules
}

# 模态组合器
structure ComposedModal<M1: Modality, M2: Modality> : Modality {
    axioms: CombinedAxioms<M1.axioms, M2.axioms>,
    transitions: CombinedRules<M1.transitions, M2.transitions>
}

# 模态规则
micro modal_intro<M: Modality, A: Type>(
    proof: A
) -> Modal<M, A>

micro modal_elim<M: Modality, A: Type>(
    modal_proof: Modal<M, A>,
    context: ModalContext<M>
) -> A

# 模态组合
micro modal_compose<M1: Modality, M2: Modality, A: Type>(
    proof: Modal<M1, Modal<M2, A>>
) -> Modal<Compose<M1, M2>, A>

# S4 模态逻辑
axiom modal_k<M: Modality, A: Type, B: Type>(
    f: Modal<M, A -> B>,
    x: Modal<M, A>
) -> Modal<M, B>

axiom modal_t<M: Modality, A: Type>(
    x: Modal<M, A>
) -> A  # 只对某些模态成立

axiom modal_4<M: Modality, A: Type>(
    x: Modal<M, A>
) -> Modal<M, Modal<M, A>>

高阶归纳类型

# 圆周类型
structure Circle : Type {
    base: Circle,
    loop: Path<Circle, base, base>
}

# 圆周递归原理
micro circle_rec<P: Type>(
    base_case: P,
    loop_case: Path<P, base_case, base_case>
) -> Circle -> P

# 圆周归纳原理
micro circle_ind<P: Circle -> Type>(
    base_case: P(Circle.base),
    loop_case: PathOver<P, Circle.loop, base_case, base_case>
) -> (c: Circle) -> P(c)

# 球面类型
structure Sphere(n: Nat) : Type {
    base: Sphere(n),
    # n维球面的生成元
    generator: Path^n<Sphere(n), base, base>
}

# 悬垂构造
structure Suspension<A: Type> : Type {
    north: Suspension<A>,
    south: Suspension<A>,
    merid: (a: A) -> Path<Suspension<A>, north, south>
}

# 推出构造
structure Pushout<A: Type, B: Type, C: Type>(
    f: A -> B, 
    g: A -> C
) : Type {
    inl: B -> Pushout<A, B, C>(f, g),
    inr: C -> Pushout<A, B, C>(f, g),
    glue: (a: A) -> Path<Pushout<A, B, C>(f, g), inl(f(a)), inr(g(a))>
}

定理证明示例

基础数学定理

# 自然数的基本性质
theorem nat_induction<P: Nat -> Prop>(
    base: P(0),
    step: (n: Nat) -> P(n) -> P(n + 1)
) -> (n: Nat) -> P(n) {
    match n {
        0 => base,
        succ(k) => step(k, nat_induction(base, step, k))
    }
}

# 加法交换律
theorem add_comm(m: Nat, n: Nat) -> Path<Nat, m + n, n + m> {
    match m {
        0 => {
            # 0 + n = n = n + 0
            rewrite {
                0 + n 
                => n              { add_zero_left }
                => n + 0          { add_zero_right.symm }
            }
        },
        succ(k) => {
            # succ(k) + n = succ(k + n) = succ(n + k) = n + succ(k)
            rewrite {
                succ(k) + n
                => succ(k + n)    { add_succ_left }
                => succ(n + k)    { ap(succ, add_comm(k, n)) }
                => n + succ(k)    { add_succ_right.symm }
            }
        }
    }
}

# 加法结合律
theorem add_assoc(a: Nat, b: Nat, c: Nat) -> Path<Nat, (a + b) + c, a + (b + c)> {
    match a {
        0 => {
            rewrite {
                (0 + b) + c
                => b + c          { ap(λx. x + c, add_zero_left) }
                => 0 + (b + c)    { add_zero_left.symm }
            }
        },
        succ(k) => {
            rewrite {
                (succ(k) + b) + c
                => succ(k + b) + c      { ap(λx. x + c, add_succ_left) }
                => succ((k + b) + c)    { add_succ_left }
                => succ(k + (b + c))    { ap(succ, add_assoc(k, b, c)) }
                => succ(k) + (b + c)    { add_succ_left.symm }
            }
        }
    }
}

群论证明

# 群的定义
structure Group {
    carrier: Type,
    op: carrier -> carrier -> carrier,
    identity: carrier,
    inverse: carrier -> carrier,
    
    # 群公理
    assoc: (a: carrier, b: carrier, c: carrier) -> 
           Path<carrier, op(op(a, b), c), op(a, op(b, c))>,
    
    left_identity: (a: carrier) -> 
                   Path<carrier, op(identity, a), a>,
    
    right_identity: (a: carrier) -> 
                    Path<carrier, op(a, identity), a>,
    
    left_inverse: (a: carrier) -> 
                  Path<carrier, op(inverse(a), a), identity>,
    
    right_inverse: (a: carrier) -> 
                   Path<carrier, op(a, inverse(a)), identity>
}

# 群同态
structure GroupHom(G: Group, H: Group) {
    map: G.carrier -> H.carrier,
    
    preserve_op: (a: G.carrier, b: G.carrier) -> 
                 Path<H.carrier, map(G.op(a, b)), H.op(map(a), map(b))>,
    
    preserve_identity: Path<H.carrier, map(G.identity), H.identity>
}

# 群同态保持逆元
theorem group_hom_preserve_inverse<G: Group, H: Group>(
    f: GroupHom(G, H),
    a: G.carrier
) -> Path<H.carrier, f.map(G.inverse(a)), H.inverse(f.map(a))> {
    # 利用逆元的唯一性
    let h1: Path<H.carrier, H.op(f.map(G.inverse(a)), f.map(a)), H.identity> = {
        rewrite {
            H.op(f.map(G.inverse(a)), f.map(a))
            => f.map(G.op(G.inverse(a), a))    { f.preserve_op.symm }
            => f.map(G.identity)               { ap(f.map, G.left_inverse(a)) }
            => H.identity                      { f.preserve_identity }
        }
    }
    
    # 由逆元的唯一性得出结论
    inverse_unique(H, f.map(a), f.map(G.inverse(a)), h1)
}

# 第一同构定理
theorem first_isomorphism_theorem<G: Group, H: Group>(
    f: GroupHom(G, H)
) -> GroupIsom(QuotientGroup(G, Kernel(f)), Image(f)) {
    # 构造同构映射
    let φ: QuotientGroup(G, Kernel(f)).carrier -> Image(f).carrier = 
        λ[g]. ⟨f.map(g.representative), image_membership(f, g.representative)⟩
    
    # 证明 φ 是良定义的
    let well_defined: (g1: QuotientGroup(G, Kernel(f)).carrier, 
                       g2: QuotientGroup(G, Kernel(f)).carrier) ->
                      Path<QuotientGroup(G, Kernel(f)).carrier, g1, g2> ->
                      Path<Image(f).carrier, φ(g1), φ(g2)> = {
        # 详细证明省略
        sorry
    }
    
    # 证明 φ 是群同态
    let is_homomorphism: GroupHom(QuotientGroup(G, Kernel(f)), Image(f)) = {
        # 详细证明省略
        sorry
    }
    
    # 证明 φ 是双射
    let is_bijective: Bijective(φ) = {
        # 详细证明省略
        sorry
    }
    
    GroupIsom {
        forward: is_homomorphism,
        backward: inverse_homomorphism(is_homomorphism, is_bijective),
        left_inverse: sorry,
        right_inverse: sorry
    }
}

拓扑学证明

# 拓扑空间
structure TopologicalSpace {
    carrier: Type,
    open_sets: Subset(PowerSet(carrier)),
    
    # 拓扑公理
    empty_open: open_sets(∅),
    total_open: open_sets(carrier),
    union_open: (family: Family(Subset(carrier))) -> 
                (∀ U ∈ family. open_sets(U)) -> 
                open_sets(⋃ family),
    intersection_open: (U: Subset(carrier), V: Subset(carrier)) ->
                       open_sets(U) -> open_sets(V) -> 
                       open_sets(U ∩ V)
}

# 连续映射
structure ContinuousMap(X: TopologicalSpace, Y: TopologicalSpace) {
    map: X.carrier -> Y.carrier,
    
    continuous: (V: Subset(Y.carrier)) -> 
                Y.open_sets(V) -> 
                X.open_sets(preimage(map, V))
}

# 同胚
structure Homeomorphism(X: TopologicalSpace, Y: TopologicalSpace) {
    forward: ContinuousMap(X, Y),
    backward: ContinuousMap(Y, X),
    
    left_inverse: (x: X.carrier) -> 
                  Path<X.carrier, backward.map(forward.map(x)), x>,
    
    right_inverse: (y: Y.carrier) -> 
                   Path<Y.carrier, forward.map(backward.map(y)), y>
}

# 基本群
structure FundamentalGroup(X: TopologicalSpace, x₀: X.carrier) {
    carrier: LoopSpace(X, x₀) / Homotopy,
    op: (α: carrier, β: carrier) -> α * β,  # 路径连接
    identity: constant_loop(x₀),
    inverse: (α: carrier) -> reverse_path(α)
}

# 范畴论中的函子
structure Functor(C: Category, D: Category) {
    object_map: C.Object -> D.Object,
    morphism_map: (A: C.Object, B: C.Object) -> 
                  C.Hom(A, B) -> D.Hom(object_map(A), object_map(B)),
    
    preserve_identity: (A: C.Object) -> 
                       Path<D.Hom(object_map(A), object_map(A)), 
                            morphism_map(A, A, C.id(A)), 
                            D.id(object_map(A))>,
    
    preserve_composition: (A: C.Object, B: C.Object, C: C.Object,
                          f: C.Hom(A, B), g: C.Hom(B, C)) ->
                         Path<D.Hom(object_map(A), object_map(C)),
                              morphism_map(A, C, C.compose(g, f)),
                              D.compose(morphism_map(B, C, g), 
                                       morphism_map(A, B, f))>
}

模态逻辑应用

认知逻辑

# 认知算子
structure Knowledge<Agent: Type, Prop: Type> {
    knows: Agent -> Prop -> Type,
    
    # 知识公理
    knowledge_implies_truth: (a: Agent, p: Prop) -> 
                            knows(a, p) -> p,
    
    positive_introspection: (a: Agent, p: Prop) -> 
                           knows(a, p) -> knows(a, knows(a, p)),
    
    negative_introspection: (a: Agent, p: Prop) -> 
                           ¬knows(a, p) -> knows(a, ¬knows(a, p))
}

# 共同知识
structure CommonKnowledge<Agents: Type, Prop: Type> {
    everyone_knows: (p: Prop) -> 
                   (∀ a: Agents. Knowledge.knows(a, p)) -> Type,
    
    common_knowledge: (p: Prop) -> Type,
    
    # 共同知识的归纳定义
    ck_base: (p: Prop) -> 
             everyone_knows(p, _) -> 
             common_knowledge(p),
    
    ck_step: (p: Prop) -> 
             common_knowledge(everyone_knows(p, _)) -> 
             common_knowledge(p)
}

# 拜占庭将军问题的形式化
theorem byzantine_generals_impossibility(
    n: Nat,
    traitors: Nat,
    assumption: traitors ≥ n / 3
) -> ¬∃(protocol: ConsensusProtocol). 
      GuaranteesConsensus(protocol, n, traitors) {
    # 反证法：假设存在这样的协议
    assume protocol: ConsensusProtocol,
           guarantee: GuaranteesConsensus(protocol, n, traitors)
    
    # 构造反例场景
    let scenario = AdversarialScenario {
        honest_generals: n - traitors,
        byzantine_generals: traitors,
        network_partition: true
    }
    
    # 证明协议在此场景下失败
    let failure: ProtocolFails(protocol, scenario) = {
        # 利用信息论论证
        information_theoretic_bound(n, traitors, assumption)
    }
    
    # 矛盾
    contradiction(guarantee.correctness(scenario), failure)
}

时态逻辑

# 线性时态逻辑 (LTL)
structure LTL<Prop: Type> {
    # 时态算子
    ○: Prop -> Prop,           # 下一个状态
    ◇: Prop -> Prop,           # 最终
    □: Prop -> Prop,           # 总是
    𝒰: Prop -> Prop -> Prop,    # 直到
    
    # 时态公理
    next_distributive: (p: Prop, q: Prop) -> 
                      Path<Prop, ○(p ∧ q), ○(p) ∧ ○(q)>,
    
    eventually_unfold: (p: Prop) -> 
                      Path<Prop, ◇(p), p ∨ ○(◇(p))>,
    
    always_unfold: (p: Prop) -> 
                  Path<Prop, □(p), p ∧ ○(□(p))>,
    
    until_unfold: (p: Prop, q: Prop) -> 
                 Path<Prop, p 𝒰 q, q ∨ (p ∧ ○(p 𝒰 q))>
}

# 计算树逻辑 (CTL)
structure CTL<Prop: Type> {
    # 路径量词
    𝒜: (Prop -> Prop) -> Prop,    # 所有路径
    ℰ: (Prop -> Prop) -> Prop,    # 存在路径
    
    # 组合算子
    𝒜□: Prop -> Prop,  # 所有路径上总是
    ℰ◇: Prop -> Prop,  # 存在路径最终
    𝒜◇: Prop -> Prop,  # 所有路径最终
    ℰ□: Prop -> Prop,  # 存在路径总是
    
    # CTL 公理
    ag_definition: (p: Prop) -> 
                  Path<Prop, 𝒜□(p), 𝒜(□(p))>,
    
    ef_definition: (p: Prop) -> 
                  Path<Prop, ℰ◇(p), ℰ(◇(p))>
}

# 模型检验定理
theorem model_checking_decidable<M: KripkeStructure, φ: CTL.Formula>() -> 
    Decidable(M ⊨ φ) {
    # 构造标记算法
    let algorithm = CTLModelCheckingAlgorithm {
        structure: M,
        formula: φ,
        
        # 自底向上标记
        label_atomic: label_atomic_propositions(M),
        label_boolean: label_boolean_combinations,
        label_temporal: label_temporal_operators
    }
    
    # 证明算法的正确性和终止性
    let correctness: AlgorithmCorrect(algorithm) = 
        structural_induction_on_formula(φ)
    
    let termination: AlgorithmTerminates(algorithm) = 
        finite_state_space_argument(M)
    
    DecidabilityProof {
        algorithm: algorithm,
        correctness: correctness,
        termination: termination
    }
}

高级证明技术

类型驱动开发

# 依赖类型的向量
structure Vec<A: Type, n: Nat> : Type {
    data: Array<A>,
    length_proof: Path<Nat, data.length, n>
}

# 类型安全的向量操作
micro vec_head<A: Type, n: Nat>(
    v: Vec<A, succ(n)>  # 非空向量
) -> A {
    v.data[0]  # 类型系统保证索引有效
}

micro vec_tail<A: Type, n: Nat>(
    v: Vec<A, succ(n)>
) -> Vec<A, n> {
    Vec {
        data: v.data[1..],
        length_proof: tail_length_correct(v)
    }
}

# 向量连接保持长度
micro vec_append<A: Type, m: Nat, n: Nat>(
    v1: Vec<A, m>,
    v2: Vec<A, n>
) -> Vec<A, m + n> {
    Vec {
        data: v1.data ++ v2.data,
        length_proof: append_length_correct(v1, v2)
    }
}

# 矩阵乘法的类型安全性
micro matrix_multiply<A: Ring, m: Nat, n: Nat, p: Nat>(
    a: Matrix<A, m, n>,
    b: Matrix<A, n, p>
) -> Matrix<A, m, p> {
    # 类型系统确保维度匹配
    Matrix {
        data: compute_matrix_product(a.data, b.data),
        dimensions_proof: multiply_dimensions_correct(a, b)
    }
}

程序验证

# 霍尔逻辑 {P} C {Q}
structure HoareTriple<State: Type> {
    𝒫: State -> Prop,  # 前置条件
    𝒞: State -> State,  # 程序
    𝒬: State -> Prop,  # 后置条件
    
    validity: (s: State) -> 𝒫(s) -> 𝒬(𝒞(s))
}

# 排序算法的正确性
theorem quicksort_correctness<A: TotalOrder>(
    arr: Array<A>
) -> HoareTriple<Array<A>> {
    HoareTriple {
        𝒫: λarr. True,  # 无前置条件
        𝒞: quicksort,
        𝒬: λresult. 
            Sorted(result) ∧ 
            Permutation(arr, result) ∧
            Path<Nat, arr.length, result.length>,
        
        validity: λs pre => match s.length {
            0 => trivial,
            1 => trivial,
            succ(succ(n)) => {
                # 分治递归情况
                let pivot = s[0]
                let (left, right) = partition(s[1..], pivot)
                
                # 归纳假设
                let ih_left = quicksort_correctness(left)
                let ih_right = quicksort_correctness(right)
                
                # 组合结果
                combine_sorted_parts(pivot, ih_left, ih_right)
            }
        }
    }
}

# 并发程序验证
structure ConcurrentProgram<State: Type> {
    processes: List<Process<State>>,
    shared_state: SharedState<State>,
    
    # 安全性属性
    safety: (s: State) -> Prop,
    
    # 活性属性
    liveness: (trace: ExecutionTrace<State>) -> Prop
}

# 互斥锁的正确性
theorem mutex_correctness(
    lock: MutexLock,
    critical_section: Process<State>
) -> ConcurrentCorrectness {
    # 安全性：互斥访问
    let mutual_exclusion: ∀(t: Time). AtMostOneProcess(InCriticalSection(t)) = 
        mutex_safety_proof(lock)
    
    # 活性：无死锁
    let deadlock_freedom: ∀(trace: ExecutionTrace). 
        ProcessWaiting(trace) → EventuallyEnters(trace) = 
        mutex_liveness_proof(lock)
    
    # 公平性：无饥饿
    let starvation_freedom: ∀(p: Process). 
        InfinitelyOftenRequests(p) → InfinitelyOftenEnters(p) = 
        mutex_fairness_proof(lock)
    
    ConcurrentCorrectness {
        safety: mutual_exclusion,
        liveness: deadlock_freedom ∧ starvation_freedom
    }
}

使用指南

基础定理证明

# 简单的命题逻辑证明
theorem modus_ponens<P: Prop, Q: Prop>(
    premise1: P,
    premise2: P -> Q
) -> Q {
    premise2(premise1)
}

# 德摩根定律证明
theorem de_morgan<P: Prop, Q: Prop>() -> 
    Path<Prop, ¬(P ∨ Q), ¬P ∧ ¬Q> {
    λ h => ⟨
        λ hp => h(Left(hp)),
        λ hq => h(Right(hq))
    ⟩
}

# 自动化证明
theorem arithmetic_example(a: Nat, b: Nat, c: Nat) -> 
    Path<Nat, (a + b) * c, a * c + b * c> {
    ring_tactic  # 自动环论证明
}

交互式证明开发

# 自然数奇偶性证明
theorem nat_even_or_odd(n: Nat) -> Even(n) ∨ Odd(n) {
    match n {
        0 => Left(even_zero),
        succ(k) => match nat_even_or_odd(k) {
            Left(h_even) => Right(odd_succ_of_even(h_even)),
            Right(h_odd) => Left(even_succ_of_odd(h_odd))
        }
    }
}

证明自动化

# 自定义策略
macro ring_solver {
    # 环论求解器
    normalize_expressions,
    apply_ring_axioms,
    simplify_arithmetic
}

macro simp_all {
    # 简化所有假设和目标
    simp at *,
    try { assumption }
}

# 决策过程
macro omega {
    # 线性算术决策过程
    presburger_arithmetic
}

# 使用自动化
theorem automated_proof(x: Int, y: Int) -> 
    Path<Int, 2 * (x + y), 2 * x + 2 * y> {
    ring_solver
}

theorem linear_arithmetic(a: Nat, b: Nat) -> 
    a < b -> a + 1 ≤ b {
    omega
}

Valkyrie 的定理证明器基于最新的类型论研究成果，提供了强大而优雅的数学证明环境。通过模态同伦类型论的基础，可以处理从基础数学到高级抽象代数、拓扑学、逻辑学等各个领域的形式化证明。