File size: 2,900 Bytes
e5a97c6 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 | import OAI.Combinatorics.PolynomialVacuum.Algebra.NaturalCandidateEndpoints
import OAI.Combinatorics.PolynomialVacuum.Algebra.FirstCupRegularNode
namespace OAI
namespace PolynomialVacuumAlgebra
open Classical
def NaturalCupNodes (Ψ ψ : NatSpectralTensor) (n : ℕ) : Prop :=
∀ X i, i ≤ n → X (n+1)=T*X i → Ψ X=naturalPrescription ψ n i X
theorem lift_interpolate_cupNodes {n : ℕ} (ψ : State n) (χ : State (n+1))
(hψ : LowerExchange (liftState ψ) n) :
NaturalCupNodes (liftState (interpolateState ψ χ)) (liftState ψ) n := by
intro X i hi hX
exact interpolateState_node_natural ψ χ hψ i hi X hX
theorem naturalFusion_first_congr (n : ℕ) (X Y : ℕ → ℂ)
(h : ∀ j < n+2, X j=Y j) : naturalFusion n 0 X=naturalFusion n 0 Y := by
simp only [naturalFusion,h 0 (by omega),fusionProduct,skipTwo_zero_apply]
congr 1
apply Finset.prod_congr rfl
intro k hk
rw [h (k+2) (by have:=Finset.mem_range.mp hk; omega)]
theorem insertOne_at_node_congr {α : Type*} (n : ℕ) (c : α) (X : ℕ → α)
(hX : X n=c) (j : ℕ) (hj : j < n+1) : insertOne n c X j=X j := by
by_cases h : j < n
· simp [insertOne,h]
· have : j=n := by omega
subst j
simp [hX]
theorem firstCup_node_equality {n : ℕ} (Ψ : NatSpectralTensor) (ψ : State (n+2))
(φ : NatSpectralTensor)
(hΨ : NaturalCupNodes Ψ (liftState ψ) (n+2))
(hψ : NaturalCupNodes (liftState ψ) φ n)
(hex : LowerExchange (liftState ψ) (n+2))
(hcup : FirstCupReduction (liftState ψ) φ n)
(X : ℕ → ℂ) (hX : X 1=T*X 0) (i : ℕ) (hi : i ≤ n+2)
(hnode : X (n+3)=T*X i) :
Ψ X=naturalFusion (n+2) 0 X • cupLinear 0 (liftState ψ (skipTwo 0 X)) := by
rw [hΨ X i hi hnode]
have hf : naturalFusion (n+2) 0 (insertOne (n+3) (T*X i) X)=naturalFusion (n+2) 0 X :=
naturalFusion_first_congr _ _ _ (insertOne_at_node_congr _ _ _ hnode)
have hp : liftState ψ (skipTwo 0 (insertOne (n+3) (T*X i) X))=liftState ψ (skipTwo 0 X) := by
apply liftState_params_congr
intro j hj
simpa using insertOne_at_node_congr (n+3) (T*X i) X hnode (j+2) (by omega)
rcases i with _|(_|j)
· rw [naturalPrescription_firstCup_regular hex X hX,hf,hp]
· rw [naturalPrescription_firstCup_middle _ _ X hX]
have hz : naturalFusion (n+2) 0 X=0 := by
simp only [naturalFusion,fusionProduct]
suffices hh : (∏ k ∈ Finset.range (n+2), fusionPair (X 0) (skipTwo 0 X k))=0 by rw [hh,mul_zero]
apply Finset.prod_eq_zero (Finset.mem_range.mpr (by omega : n+1 < n+2))
simpa only [skipTwo_zero_apply,show n+1+2=n+3 by omega,hnode,hX] using fusionPair_second_successor_zero (X 0)
rw [hz,zero_smul]
· rw [naturalPrescription_firstCup_after hcup j (by omega) X hX,hf]
rw [hψ (skipTwo 0 X) j (by omega) (by simpa only [skipTwo_zero_apply,show n+1+2=n+3 by omega] using hnode)]
end PolynomialVacuumAlgebra
end OAI
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