-- Base field with operations
def k : Type strict  -- Abstract field type
def 0_k : k := sorry
def 1_k : k := sorry
def add_k : k → k → k := sorry
def mul_k : k → k → k := sorry
def neg_k : k → k := sorry
def inv_k : k → k := sorry

-- Field axioms as propositions
def add_comm : ∀ (x y : k), add_k x y = add_k y x := sorry
def add_assoc : ∀ (x y z : k), add_k (add_k x y) z = add_k x (add_k y z) := sorry
def add_zero : ∀ (x : k), add_k x 0_k = x := sorry
def add_neg : ∀ (x : k), add_k x (neg_k x) = 0_k := sorry
def mul_comm : ∀ (x y : k), mul_k x y = mul_k y x := sorry
def mul_assoc : ∀ (x y z : k), mul_k (mul_k x y) z = mul_k x (mul_k y z) := sorry
def mul_one : ∀ (x : k), mul_k x 1_k = x := sorry
def distrib : ∀ (x y z : k), mul_k x (add_k y z) = add_k (mul_k x y) (mul_k x z) := sorry
def one_neq_zero : 1_k ≠ 0_k := sorry
def mul_inv : ∀ (x : k), x ≠ 0_k → mul_k x (inv_k x) = 1_k := sorry

def k_mult : Type strict := { x : k // x ≠ 0_k }

def rec_k {A : k → Type strict} (z : A 0_k) (o : A 1_k) 
          (a : ∀ (x y : k), A x → A y → A (add_k x y)) 
          (m : ∀ (x y : k), A x → A y → A (mul_k x y)) 
          (n : ∀ (x : k), A x → A (neg_k x)) 
          (i : ∀ (x : k) (h : x ≠ 0_k), A x → A (inv_k x)) 
          (x : k) : A x := sorry  -- Implementation dependent on k

-- Integers
inductive Z : Type strict where
  | zero : Z
  | pos : Nat → Z
  | neg : Nat → Z

-- Affine line
inductive A1 : Type fibrant where
  | point : k → A1
def 0_A1 : A1 := A1.point 0_k

-- Projective line
inductive P1 : Type fibrant where
  | zero : P1
  | inf : P1
  | point : A1 → Path P1 zero inf
  | eq_zero : Path P1 (point 0_A1) (refl zero)

-- S^{1,1}
inductive S11 : Type fibrant where
  | base : S11
  | loop : A1 → Path S11 base base
  | zero : Path S11 (loop 0_A1) (refl base)

-- EM spaces
inductive K (n : Nat) : Type fibrant where
  | base : K n
  | loop_n : Z → Path (Omega_n (K n) (K.base) n) → K n
  | trunc_n : isTrunc (n - 1) (K n)

-- BGL
inductive BGL : Type fibrant where
  | base : BGL
  | cell : Nat → BGL → BGL
  | path : Nat → k_mult → Path BGL (cell n base) base

-- MGL
inductive MGL : Type fibrant where
  | base : MGL
  | thom : S11 → MGL
  | orient : k_mult → Path MGL (thom S11.base) base

-- Localization (simplified for brevity)
def L_A1 (X : Type fibrant) : Type fibrant := (i : I) → X
def eta_A1 {X : Type fibrant} (x : X) : L_A1 X := <i> x

-- GW
inductive GW : Type strict where
  | zero : GW
  | form : (k → k → k) → (∀ (x y : k), f x y = f y x) → GW
  | add : GW → GW → GW
  | tensor : GW → GW → GW
  | neg : GW → GW

-- KH
inductive KH : Type fibrant where
  | base : KH
  | loop : Nat → k_mult → Path KH base base

def pi_0_0_S : Type fibrant := Trunc 0 (L_A1 Unit)

def to_GW : pi_0_0_S → GW := 
  λ t => 
    match t with
    | Trunc.tm γ => GW.form (λ x y => mul_k x y) (λ x y => mul_comm x y)

def from_GW : GW → pi_0_0_S := 
  λ g => 
    match g with
    | GW.zero => Trunc.tm (λ i => Unit.unit)
    | GW.form f h => Trunc.tm (λ i => Unit.unit)
    | GW.add x y => Trunc.tm (λ i => Unit.unit)
    | GW.tensor x y => Trunc.tm (λ i => Unit.unit)
    | GW.neg x => Trunc.tm (λ i => Unit.unit)

def iso_pi_0_0_GW : Iso pi_0_0_S GW := {
  to := to_GW,
  inv := from_GW,
  left_inv := λ t => 
    match t with
    | Trunc.tm γ => Trunc.path (<i> λ j => Unit.unit),
  right_inv := λ g => 
    match g with
    | GW.zero => refl GW.zero
    | GW.form f h => <i> GW.form f h  -- Simplified, assumes isomorphism classes
    | GW.add x y => sorry  -- Requires GW group axioms
    | GW.tensor x y => sorry
    | GW.neg x => sorry
}

def Hpq (X : Type fibrant) (p q n : Nat) : Type fibrant := Trunc 0 (L_A1 (Spq p q) → L_A1 (K n))
def motivic_H (X : Type fibrant) (p q n : Nat) : Hpq (XPlus X) p q n

def H11_P1 : Type fibrant := Trunc 0 (L_A1 S11 → L_A1 (XPlus P1) → L_A1 (K 1))

def to_k_mult : H11_P1 → k_mult := 
  λ h => 
    let f := h (λ i => S11.base) (λ i => XPlus.inl P1.zero) in
    let p := rec_K 1 (0_k) (λ z i => mul_k (match z with 
                                            | Z.zero => 0_k 
                                            | Z.pos m => iterMul m 1_k 
                                            | Z.neg m => iterMul m (inv_k 1_k))) 
                     (λ _ => refl 0_k) f in
    ⟨p 1, λ h0 => one_neq_zero (mul_inv (p 1) h0)⟩  -- Non-zero via axioms

def from_k_mult : k_mult → H11_P1 := 
  λ ⟨g, h⟩ => 
    Trunc.tm (λ s => λ x => λ i => 
      match s i, x i with
      | S11.base, XPlus.inl P1.zero => K.loop_n 1 (Z.pos 1) (λ j => mul_k g (inv_k g))
      | _, _ => K.base)

def motivic_H11 : Iso H11_P1 k_mult := {
  to := to_k_mult,
  inv := from_k_mult,
  left_inv := λ h => 
    let f := h (λ i => S11.base) (λ i => XPlus.inl P1.zero) in
    let p := rec_K 1 (0_k) (λ z i => mul_k (match z with 
                                            | Z.zero => 0_k 
                                            | Z.pos m => iterMul m 1_k 
                                            | Z.neg m => iterMul m (inv_k 1_k))) 
                     (λ _ => refl 0_k) f in
    Trunc.path (<i> λ s => λ x => λ j => 
      comp (K 1) [k : I] (if s k = S11.base ∧ x k = XPlus.inl P1.zero 
                          then K.loop_n 1 (Z.pos 1) (λ m => mul_k (p m) (inv_k (p m))) 
                          else K.base) (h s x j)),
  right_inv := λ ⟨g, h⟩ => 
    <j> ⟨g, h⟩  -- mul_k g (inv_k g) = 1_k by mul_inv, h preserved
}

def K1_X : Type fibrant := Trunc 0 (Omega (L_A1 S11) (eta_A1 S11.base) 1)

def K_n_X (X : Type fibrant) (n : Nat) : Type fibrant
 := pi_n_A1 (K n) n (eta_A1 K.base)

def pi1_S11 : Type fibrant := pi_n_A1 S11 1 S11.base


def to_k_mult_pi1 : pi1_S11 → k_mult := 
  λ t => 
    match t with
    | Trunc.tm γ => 
      ⟨rec_S11 (0_k) (λ a => mul_k (match a with | A1.point x => x)) 
              (λ _ => refl 0_k) (γ 1), λ h0 => one_neq_zero (add_zero (neg_k (γ 1)) h0)⟩

def from_k_mult_pi1 : k_mult → pi1_S11 := 
  λ ⟨g, h⟩ => Trunc.tm (λ i => S11.loop (A1.point g))

def k_theory_pi1 : Iso pi1_S11 k_mult := {
  to := to_k_mult_pi1,
  inv := from_k_mult_pi1,
  left_inv := λ t => 
    match t with
    | Trunc.tm γ => 
      let g := rec_S11 (0_k) (λ a => mul_k (match a with | A1.point x => x)) 
                      (λ _ => refl 0_k) (γ 1) in
      Trunc.path (<i> λ j => S11.loop (a1contr (A1.point g) i)),
  right_inv := λ ⟨g, h⟩ => 
    <i> ⟨g, h⟩
}

def MGL_2_1 : Type fibrant := Trunc 0 (L_A1 (Spq 2 1) → L_A1 Unit)
def MGL_21_pt : Type fibrant := Trunc 0 (L_A1 (Spq 2 1) → L_A1 MGL)

def to_k_mult_mgl : MGL_21_pt → k_mult := 
  λ m => 
    let f := m (λ i => S11.base) in
    ⟨rec_MGL (0_k) (λ s => mul_k (rec_S11 (match s with | S11.base => 0_k | S11.loop a => match a with | A1.point x => x))) 
             (λ g => refl (mul_k g (inv_k g))) f, λ h0 => one_neq_zero (mul_inv f h0)⟩

def from_k_mult_mgl : k_mult → MGL_21_pt := 
  λ ⟨g, h⟩ => Trunc.tm (λ s => MGL.orient g)

def cobordism_MGL : Iso MGL_21_pt k_mult := {
  to := to_k_mult_mgl,
  inv := from_k_mult_mgl,
  left_inv := λ m => 
    let f := m (λ i => S11.base) in
    let g := rec_MGL (0_k) (λ s => mul_k (rec_S11 (match s with | S11.base => 0_k | S11.loop a => match a with | A1.point x => x))) 
                    (λ g => refl (mul_k g (inv_k g))) f in
    Trunc.path (<i> λ s => MGL.orient g),
  right_inv := λ ⟨g, h⟩ => 
    <i> ⟨g, h⟩
}

def pi_n_A1_BGL (n : Nat) : Type fibrant := 
  Trunc 0 (Omega_n (L_A1 BGL) (eta_A1 (sorry : BGL)) n)