openAI-math-dataset / lean /OAI /Combinatorics /PolynomialVacuum /Algebra /FirstCupNodeAssembly.lean
| 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 | |