| import Mathlib |
| import OAI.Algebra.FiniteTensor.CompatibleCharts |
|
|
| namespace OAI |
|
|
| /-! Positive-characteristic tensor obstruction and exact formal parametrization conditions. -/ |
|
|
| noncomputable section |
| open scoped BigOperators |
|
|
| namespace PD4Tensor.TruncatedForms |
| noncomputable section |
| open scoped BigOperators |
| open FrobeniusTruncation FiniteCoordinates |
| attribute [local instance] parameterCharP |
| universe u |
| variable (K : Type*) [Field K] (p : ℕ) [CharP K p] [Fact p.Prime] [Invertible (2 : K)] |
| {n : ℕ} (σ : Fin (n+1) → Type u) [∀ i,Fintype (σ i)] [∀ i,DecidableEq (σ i)] |
|
|
| include p in |
| |
|
|
| theorem formal_positive_obstruction {d c : ℕ} |
| (e : ((i : Fin (n+1)) × σ i) ≃ Fin d ⊕ Fin c) |
| (F b : ∀ i,MvPowerSeries (σ i) K) (active : Finset (Fin (n+1))) |
| (hb : ∀ i,i∉active → b i=0) (hA : 5≤active.card) |
| (H0 : ((i : Fin (n+1)) × σ i) → MvPowerSeries (Fin d) K) |
| (G : ((i : Fin (n+1)) × σ i) → MvPowerSeries (Fin c) K) |
| (H : (Fin 3 ↪ Fin (n+1)) → ((i : Fin (n+1)) × σ i) → MvPowerSeries (Fin d) (T K 3)) |
| (hH0z : ∀ i,MvPowerSeries.constantCoeff (H0 i)=0) |
| (hGz : ∀ i,MvPowerSeries.constantCoeff (G i)=0) |
| (hH : ∀ q i,MvPowerSeries.map (augmentation K 3) (H q i)=H0 i) |
| (hHL : Function.Injective (Matrix.mulVecLin |
| (fun i j => MvPowerSeries.coeff (Finsupp.single j 1) (H0 i)))) |
| (hGR : Function.Injective (Matrix.mulVecLin |
| (fun i j => MvPowerSeries.coeff (Finsupp.single j 1) (G i)))) |
| (hcomp : IsCompl (Matrix.mulVecLin |
| (fun i j => MvPowerSeries.coeff (Finsupp.single j 1) (H0 i))).range |
| (Matrix.mulVecLin (fun i j => MvPowerSeries.coeff (Finsupp.single j 1) (G i))).range) |
| (hL0 : MvPowerSeries.subst H0 (formalSigmaPotential K σ F)=0) |
| (hR : MvPowerSeries.subst G (formalSigmaPotential K σ F)=0) |
| (htripR : ∀ s : Finset (Fin (n+1)), s⊆active → s.card=3 → |
| MvPowerSeries.subst G (formalSigmaSelected K σ b s)∈Ideal.span |
| (Set.range (fun i => MvPowerSeries.subst G |
| (MvPowerSeries.pderiv i (formalSigmaPotential K σ F))))) |
| (hL : ∀ q : Fin 3 ↪ Fin (n+1), (∀ j,q j∈active) → |
| MvPowerSeries.subst (H q) (formalSigmaDeformed K σ F b q)=0) |
| (htripL : ∀ q : Fin 3 ↪ Fin (n+1), (∀ j,q j∈active) → |
| MvPowerSeries.C (t K 3 0*t K 3 1*t K 3 2)∈Ideal.span |
| (Set.range (fun i => MvPowerSeries.subst (H q) |
| (MvPowerSeries.pderiv i (formalSigmaDeformed K σ F b q))))) : False := by |
| classical |
| let : CharP (T K 3) p := parameterCharP K 3 p |
| let : Algebra K (T K 3) := inferInstance |
| obtain ⟨q0,hq0L,hq0R,hqs⟩ := compatible_graph_charts K _ (Fin d) (Fin c) p |
| (Fin 3) (fun _ => 2) (fun _ => by decide) (fun _ => (Fact.out : p.Prime).two_le) |
| e H0 G hH0z hGz hHL hGR hcomp |
| have hHc (q : Fin 3 ↪ Fin (n+1)) (i) : |
| MvPowerSeries.map (constantMap K (Fin 3) (fun _ => 2) (fun _ => by decide)).toRingHom |
| (H q i)=H0 i := by |
| rw [constantMap_parameters] |
| exact hH q i |
| choose qL hqLL hqSS using fun q => hqs (H q) (hHc q) |
| let r := (e.trans finSumFinEquiv).trans (finCongr (Nat.add_comm d c)) |
| have hH0sub := MvPowerSeries.hasSubst_of_constantCoeff_zero hH0z |
| have hGsub := MvPowerSeries.hasSubst_of_constantCoeff_zero hGz |
| have hHsub (q : Fin 3 ↪ Fin (n+1)) : MvPowerSeries.HasSubst (H q) := by |
| apply parameter_hasSubst |
| intro i |
| exact (congrArg MvPowerSeries.constantCoeff (hH q i)).trans (hH0z i) |
| apply finite_chart_obstruction K p σ r e q0 qL |
| (fun i => formalEval K (σ i) p (F i)) (fun i => formalEval K (σ i) p (b i)) |
| active (fun i hi => by rw [hb i hi,map_zero]) hA |
| · rw [←formalSigmaPotential_eval] |
| exact graph_zero K _ (Fin d) (Fin c) p (Fact.out : p.Prime).pos e q0 |
| H0 hH0sub hq0L _ hL0 |
| · rw [←formalSigmaPotential_eval] |
| exact graph_zero K _ (Fin c) (Fin d) p (Fact.out : p.Prime).pos |
| (e.trans (Equiv.sumComm _ _)) q0 G hGsub hq0R _ hR |
| · intro s hs hc |
| rw [←formalSigmaSelected_eval] |
| simp only [←formalSigmaPotential_eval] |
| exact graph_gradient K K _ (Fin c) (Fin d) p (Fact.out : p.Prime).pos |
| (e.trans (Equiv.sumComm _ _)) q0 G hGsub hq0R |
| (formalSigmaPotential K σ F) (formalSigmaSelected K σ b s) (htripR s hs hc) |
| · intro q hq a |
| have hh := hqSS q a |
| rw [constantMap_parameters] at hh |
| exact hh |
| · intro q hq |
| rw [←formalSigmaDeformed_eval] |
| exact graph_zero (T K 3) ((i : Fin (n+1)) × σ i) (Fin d) (Fin c) |
| p (Fact.out : p.Prime).pos |
| e (qL q) (H q) (hHsub q) (hqLL q) _ (hL q hq) |
| · intro q hq |
| rw [←formalSigmaDeformed_eval] |
| have ht : MvPowerSeries.subst (H q) |
| (MvPowerSeries.C (t K 3 0*t K 3 1*t K 3 2))∈Ideal.span |
| (Set.range (fun i => MvPowerSeries.subst (H q) |
| (MvPowerSeries.pderiv i (formalSigmaDeformed K σ F b q)))) := by |
| simpa only [MvPowerSeries.subst_C] using htripL q hq |
| have hh := @graph_gradient K (T K 3) ((i : Fin (n+1)) × σ i) (Fin d) (Fin c) |
| inferInstance inferInstance (inferInstance : Algebra K (T K 3)) |
| inferInstance inferInstance inferInstance p (parameterCharP K 3 p) (Fact.out : p.Prime).pos |
| e (qL q) (H q) (hHsub q) (hqLL q) (formalSigmaDeformed K σ F b q) |
| (MvPowerSeries.C (t K 3 0*t K 3 1*t K 3 2)) ht |
| erw [show nilEval (T K 3) (coord (T K 3) ((i : Fin (n+1)) × σ i) (fun _ => p)) (fun j => ⟨p,coord_pow _ _ _ j⟩) (MvPowerSeries.C (t K 3 0*t K 3 1*t K 3 2))= algebraMap (T K 3) (Ring (T K 3) ((i : Fin (n+1)) × σ i) (fun _ => p)) (t K 3 0*t K 3 1*t K 3 2) from (nilEval (T K 3) _ _).commutes _] at hh |
| exact hh |
|
|
| end |
| end PD4Tensor.TruncatedForms |
|
|
| namespace PD4Tensor.TruncatedForms |
| noncomputable section |
| open scoped BigOperators |
| universe u |
| variable (K : Type*) [Field K] (p : ℕ) [CharP K p] [Fact p.Prime] [Invertible (2 : K)] |
| {n : ℕ} (σ : Fin (n+1) → Type u) [∀ i,Fintype (σ i)] [∀ i,DecidableEq (σ i)] |
|
|
| include p in |
| |
|
|
| theorem formal_active_obstruction {d c m : ℕ} |
| (e : ((i : Fin (n+1)) × σ i) ≃ Fin d ⊕ Fin c) |
| (hm : 5 ≤ m) (F : ∀ i,MvPowerSeries (σ i) K) |
| (active : Fin m ↪ Fin (n+1)) (b : ∀ j,MvPowerSeries (σ (active j)) K) |
| (H : ((i : Fin (n+1)) × σ i) → MvPowerSeries (Fin d) (T K m)) |
| (G : ((i : Fin (n+1)) × σ i) → MvPowerSeries (Fin c) K) |
| (hHz : ∀ i,MvPowerSeries.constantCoeff (H i)∈parameterIdeal K m) |
| (hGz : ∀ i,MvPowerSeries.constantCoeff (G i)=0) |
| (hHL : Function.Injective (Matrix.mulVecLin |
| (fun i j => augmentation K m (MvPowerSeries.coeff (Finsupp.single j 1) (H i))))) |
| (hGR : Function.Injective (Matrix.mulVecLin |
| (fun i j => MvPowerSeries.coeff (Finsupp.single j 1) (G i)))) |
| (hcomp : IsCompl (Matrix.mulVecLin |
| (fun i j => augmentation K m (MvPowerSeries.coeff (Finsupp.single j 1) (H i)))).range |
| (Matrix.mulVecLin (fun i j => MvPowerSeries.coeff (Finsupp.single j 1) (G i))).range) |
| (hL : MvPowerSeries.subst H (formalActiveDeformed K σ m F active b)=0) |
| (hR : MvPowerSeries.subst G (formalSigmaPotential K σ F)=0) |
| (htripL : ∀ i j k : Fin m,i≠j → i≠k → j≠k → |
| MvPowerSeries.C (t K m i*t K m j*t K m k)∈Ideal.span |
| (Set.range (fun x => MvPowerSeries.subst H |
| (MvPowerSeries.pderiv x (formalActiveDeformed K σ m F active b))))) |
| (htripR : ∀ i j k : Fin m,i≠j → i≠k → j≠k → |
| MvPowerSeries.subst G |
| (MvPowerSeries.rename (Sigma.mk (active i)) (b i)* |
| MvPowerSeries.rename (Sigma.mk (active j)) (b j)* |
| MvPowerSeries.rename (Sigma.mk (active k)) (b k))∈Ideal.span |
| (Set.range (fun x => MvPowerSeries.subst G |
| (MvPowerSeries.pderiv x (formalSigmaPotential K σ F))))) : False := by |
| classical |
| let A : Finset (Fin (n+1)) := Finset.univ.map active |
| let H0 := fun i => MvPowerSeries.map (augmentation K m) (H i) |
| let Hq := fun (q : Fin 3 ↪ Fin (n+1)) i => |
| MvPowerSeries.map (parameterSelector K m active q).toRingHom (H i) |
| have hHsub : MvPowerSeries.HasSubst H := |
| parameter_hasSubst K m H (fun i => augmentation_zero_of_mem K m _ (hHz i)) |
| have hmem (i : Fin (n+1)) : i∈A ↔ i∈Set.range active := by |
| simp only [A,Finset.mem_map,Finset.mem_univ,true_and,Set.mem_range] |
| apply formal_positive_obstruction K p σ e F (activeExtension active b) A |
| (fun i hi => activeExtension_other (A:=fun i => MvPowerSeries (σ i) K) |
| active b i (fun h => hi ((hmem i).mpr h))) |
| (by simpa only [A,Finset.card_map,Finset.card_univ,Fintype.card_fin] using hm) H0 G Hq |
| · intro i |
| change MvPowerSeries.constantCoeff (MvPowerSeries.map (augmentation K m) (H i))=0 |
| rw [MvPowerSeries.constantCoeff_map] |
| exact augmentation_zero_of_mem K m _ (hHz i) |
| · exact hGz |
| · intro q i |
| exact augmentation_selector_graph K m active q H i |
| · simpa only [H0,MvPowerSeries.coeff_map] using hHL |
| · exact hGR |
| · simpa only [H0,MvPowerSeries.coeff_map] using hcomp |
| · exact central_graph_zero K σ m F active b H hHsub hL |
| · exact hR |
| · intro s hs hs3 |
| exact active_right_triples K σ m F active b G htripR s |
| (fun i hi => (hmem i).mp (hs hi)) hs3 |
| · intro q _ |
| exact selector_graph_zero K σ m F active b H hHsub hL q |
| · intro q hq |
| exact selector_graph_triple K σ m F active b H hHsub htripL q |
| (fun j => (hmem (q j)).mp (hq j)) |
|
|
| end |
| end PD4Tensor.TruncatedForms |
|
|
| namespace PD4Tensor.TruncatedForms |
| noncomputable section |
| open scoped BigOperators |
| universe u |
| variable (K : Type*) [Field K] {l : ℕ} (σ : Fin l → Type u) |
| [∀ i,Fintype (σ i)] [∀ i,DecidableEq (σ i)] |
|
|
| def formalAnyPotential (F : ∀ i,MvPowerSeries (σ i) K) : |
| MvPowerSeries ((i : Fin l) × σ i) K := |
| ∑ i,MvPowerSeries.rename (Sigma.mk i) (F i) |
|
|
| def formalAnyDeformed (m : ℕ) (F : ∀ i,MvPowerSeries (σ i) K) |
| (active : Fin m ↪ Fin l) (b : ∀ j,MvPowerSeries (σ (active j)) K) : |
| MvPowerSeries ((i : Fin l) × σ i) (T K m) := |
| MvPowerSeries.map (algebraMap K (T K m)) (formalAnyPotential K σ F) + |
| ∑ j,MvPowerSeries.C (t K m j)*MvPowerSeries.map (algebraMap K (T K m)) |
| (MvPowerSeries.rename (Sigma.mk (active j)) (b j)) |
|
|
| variable (p : ℕ) [CharP K p] [Fact p.Prime] [Invertible (2 : K)] |
|
|
| include p in |
| |
| theorem formal_active_obstruction_any {d c m : ℕ} |
| (e : ((i : Fin l) × σ i) ≃ Fin d ⊕ Fin c) |
| (hm : 5 ≤ m) (F : ∀ i,MvPowerSeries (σ i) K) |
| (active : Fin m ↪ Fin l) (b : ∀ j,MvPowerSeries (σ (active j)) K) |
| (H : ((i : Fin l) × σ i) → MvPowerSeries (Fin d) (T K m)) |
| (G : ((i : Fin l) × σ i) → MvPowerSeries (Fin c) K) |
| (hHz : ∀ i,MvPowerSeries.constantCoeff (H i)∈parameterIdeal K m) |
| (hGz : ∀ i,MvPowerSeries.constantCoeff (G i)=0) |
| (hHL : Function.Injective (Matrix.mulVecLin |
| (fun i j => augmentation K m (MvPowerSeries.coeff (Finsupp.single j 1) (H i))))) |
| (hGR : Function.Injective (Matrix.mulVecLin |
| (fun i j => MvPowerSeries.coeff (Finsupp.single j 1) (G i)))) |
| (hcomp : IsCompl (Matrix.mulVecLin |
| (fun i j => augmentation K m (MvPowerSeries.coeff (Finsupp.single j 1) (H i)))).range |
| (Matrix.mulVecLin (fun i j => MvPowerSeries.coeff (Finsupp.single j 1) (G i))).range) |
| (hL : MvPowerSeries.subst H (formalAnyDeformed K σ m F active b)=0) |
| (hR : MvPowerSeries.subst G (formalAnyPotential K σ F)=0) |
| (htripL : ∀ i j k : Fin m,i≠j → i≠k → j≠k → |
| MvPowerSeries.C (t K m i*t K m j*t K m k)∈Ideal.span |
| (Set.range (fun x => MvPowerSeries.subst H |
| (MvPowerSeries.pderiv x (formalAnyDeformed K σ m F active b))))) |
| (htripR : ∀ i j k : Fin m,i≠j → i≠k → j≠k → |
| MvPowerSeries.subst G |
| (MvPowerSeries.rename (Sigma.mk (active i)) (b i)* |
| MvPowerSeries.rename (Sigma.mk (active j)) (b j)* |
| MvPowerSeries.rename (Sigma.mk (active k)) (b k))∈Ideal.span |
| (Set.range (fun x => MvPowerSeries.subst G |
| (MvPowerSeries.pderiv x (formalAnyPotential K σ F))))) : False := by |
| cases l with |
| | zero => exact Fin.elim0 (active ⟨0, by omega⟩) |
| | succ n => |
| exact formal_active_obstruction K p σ e hm F active b H G hHz hGz hHL hGR hcomp |
| hL hR htripL htripR |
|
|
| end |
| end PD4Tensor.TruncatedForms |
|
|
| namespace PD4Tensor |
|
|
| open scoped BigOperators |
|
|
| variable (K : Type*) [Field K] (m : ℕ) |
|
|
| |
| |
| abbrev Coordinates (n : Fin (2*m) → ℕ) := (i : Fin (2*m)) × Fin (n i) |
|
|
| abbrev N (n : Fin (2*m) → ℕ) := Fintype.card (Coordinates m n) |
|
|
| |
| def includeBlock {n : Fin (2*m) → ℕ} (i : Fin (2*m)) |
| (F : MvPowerSeries (Fin (n i)) K) : MvPowerSeries (Coordinates m n) K := |
| MvPowerSeries.rename (fun j => (⟨i, j⟩ : Coordinates m n)) F |
|
|
| |
| structure Germs (n : Fin (2*m) → ℕ) (active : Fin m ↪ Fin (2*m)) where |
| F : (i : Fin (2*m)) → MvPowerSeries (Fin (n i)) K |
| b : (i : Fin m) → MvPowerSeries (Fin (n (active i))) K |
| F_algebraic : ∀ i, IsAlgebraic (MvPolynomial (Fin (n i)) K) (F i) |
| b_algebraic : ∀ i, IsAlgebraic (MvPolynomial (Fin (n (active i))) K) (b i) |
| F_zero : ∀ i, MvPowerSeries.constantCoeff (F i) = 0 |
| b_zero : ∀ i, MvPowerSeries.constantCoeff (b i) = 0 |
|
|
| variable {K m} |
|
|
| def Germs.S₀ {n : Fin (2*m) → ℕ} {active : Fin m ↪ Fin (2*m)} |
| (g : Germs K m n active) : MvPowerSeries (Coordinates m n) K := |
| ∑ i, includeBlock K m i (g.F i) |
|
|
| def Germs.B {n : Fin (2*m) → ℕ} {active : Fin m ↪ Fin (2*m)} |
| (g : Germs K m n active) (i : Fin m) : MvPowerSeries (Coordinates m n) K := |
| includeBlock K m (active i) (g.b i) |
|
|
| def Germs.Sₜ {n : Fin (2*m) → ℕ} {active : Fin m ↪ Fin (2*m)} |
| (g : Germs K m n active) : MvPowerSeries (Coordinates m n) (T K m) := |
| MvPowerSeries.map (algebraMap K (T K m)) g.S₀ + |
| ∑ i, MvPowerSeries.C (t K m i) * MvPowerSeries.map (algebraMap K (T K m)) (g.B i) |
|
|
| variable (K m) |
|
|
| |
|
|
| structure Parametrizations (n : Fin (2*m) → ℕ) (d : ℕ) where |
| dimension_le : d ≤ N m n |
| HL : Coordinates m n → MvPowerSeries (Fin d) (T K m) |
| HR : Coordinates m n → MvPowerSeries (Fin (N m n - d)) K |
| HL_zero : ∀ ν, MvPowerSeries.constantCoeff (HL ν) ∈ parameterIdeal K m |
| HR_zero : ∀ ν, MvPowerSeries.constantCoeff (HR ν) = 0 |
|
|
| variable {K m} |
|
|
| def Parametrizations.leftTangent {n : Fin (2*m) → ℕ} {d : ℕ} |
| (h : Parametrizations K m n d) : (Fin d → K) →ₗ[K] Coordinates m n → K := |
| Matrix.mulVecLin (fun ν j => augmentation K m (MvPowerSeries.coeff (Finsupp.single j 1) (h.HL ν))) |
|
|
| def Parametrizations.rightTangent {n : Fin (2*m) → ℕ} {d : ℕ} |
| (h : Parametrizations K m n d) : (Fin (N m n - d) → K) →ₗ[K] Coordinates m n → K := |
| Matrix.mulVecLin (fun ν j => MvPowerSeries.coeff (Finsupp.single j 1) (h.HR ν)) |
|
|
| def Germs.leftJacobian {n : Fin (2*m) → ℕ} {active : Fin m ↪ Fin (2*m)} |
| (g : Germs K m n active) {d : ℕ} (h : Parametrizations K m n d) : |
| Ideal (MvPowerSeries (Fin d) (T K m)) := |
| Ideal.span (Set.range (fun ν => |
| MvPowerSeries.subst h.HL (MvPowerSeries.pderiv ν g.Sₜ))) |
|
|
| def Germs.rightJacobian {n : Fin (2*m) → ℕ} {active : Fin m ↪ Fin (2*m)} |
| (g : Germs K m n active) {d : ℕ} (h : Parametrizations K m n d) : |
| Ideal (MvPowerSeries (Fin (N m n - d)) K) := |
| Ideal.span (Set.range (fun ν => |
| MvPowerSeries.subst h.HR (MvPowerSeries.pderiv ν g.S₀))) |
|
|
| |
|
|
| structure TensorConditions {n : Fin (2*m) → ℕ} {active : Fin m ↪ Fin (2*m)} |
| (g : Germs K m n active) {d : ℕ} (h : Parametrizations K m n d) : Prop where |
| left_injective : Function.Injective h.leftTangent |
| right_injective : Function.Injective h.rightTangent |
| complementary : IsCompl (LinearMap.range h.leftTangent) (LinearMap.range h.rightTangent) |
| left_vanishing : MvPowerSeries.subst h.HL g.Sₜ = 0 |
| right_vanishing : MvPowerSeries.subst h.HR g.S₀ = 0 |
| left_triple : ∀ i j k : Fin m, i ≠ j → i ≠ k → j ≠ k → |
| MvPowerSeries.C (t K m i * t K m j * t K m k) ∈ g.leftJacobian h |
| right_triple : ∀ i j k : Fin m, i ≠ j → i ≠ k → j ≠ k → |
| MvPowerSeries.subst h.HR (g.B i * g.B j * g.B k) ∈ g.rightJacobian h |
|
|
| |
| def FiniteTensorObstruction : Prop := |
| ∀ (K : Type) (_ : Field K) (_ : CharZero K) (m : ℕ), 5 ≤ m → |
| ∀ (n : Fin (2*m) → ℕ) (active : Fin m ↪ Fin (2*m)) |
| (g : Germs K m n active) (d : ℕ) (h : Parametrizations K m n d), |
| ¬ TensorConditions g h |
|
|
| end PD4Tensor |
|
|
| namespace PD4Tensor |
| noncomputable section |
| open scoped BigOperators |
| variable (K : Type*) [Field K] (p : ℕ) [CharP K p] [Fact p.Prime] |
|
|
| include p in |
| |
|
|
| theorem positiveTensorObstruction (hp : p ≠ 2) (m : ℕ) (hm : 5 ≤ m) |
| (n : Fin (2*m) → ℕ) (active : Fin m ↪ Fin (2*m)) |
| (g : Germs K m n active) (d : ℕ) (h : Parametrizations K m n d) : |
| ¬ TensorConditions g h := by |
| classical |
| intro hc |
| have h2 : (2 : K) ≠ 0 := by |
| intro hz |
| have hd : p ∣ 2 := (CharP.cast_eq_zero_iff K p 2).mp hz |
| rcases (Nat.dvd_prime Nat.prime_two).mp hd with h1 | h2 |
| · exact (Fact.out : p.Prime).ne_one h1 |
| · exact hp h2 |
| let : Invertible (2 : K) := invertibleOfNonzero h2 |
| let e : Coordinates m n ≃ Fin d ⊕ Fin (N m n-d) := |
| Fintype.equivOfCardEq (by |
| simp only [Fintype.card_sum,Fintype.card_fin] |
| exact (Nat.add_sub_cancel' h.dimension_le).symm) |
| exact TruncatedForms.formal_active_obstruction_any K (fun i => Fin (n i)) p e |
| hm g.F active g.b h.HL h.HR h.HL_zero h.HR_zero hc.left_injective |
| hc.right_injective hc.complementary hc.left_vanishing hc.right_vanishing |
| hc.left_triple hc.right_triple |
| |
| end |
| end PD4Tensor |
| end |
| |
| end OAI |
| |
