| import OAI.Combinatorics.PolynomialVacuum.Algebra.BaseNatural |
| import OAI.Combinatorics.PolynomialVacuum.Algebra.BaseThreeNode |
| |
| namespace OAI |
|
|
| namespace PolynomialVacuumAlgebra |
| open MvPolynomial Classical |
|
|
| theorem baseThree_lastCup (x y : ℂ) (a b c : Spin) : |
| baseThree x y (T*y) a b c = |
| (1+T)*y * (x-t^5*y)*(x-t^6*y) * cup (b,c) * (if a=Spin.zero then 1 else 0) := by |
| rw [← baseThree_rotation y (T*y) x a b c,baseThree_cup] |
| cases a < |
|
|
| theorem vacuumThree_lastCup : CupAt (liftState vacuumThree) (liftState vacuumOne) 1 1 := by |
| intro X hX |
| funext s |
| simpa [naturalFusion,fusionProduct,fusionPair,cupLinear,skipTwo,hX,mul_assoc] using baseThree_lastCup (X 0) (X 1) (s 0) (s 1) (s 2) |
|
|
| theorem vacuumTwo_nodes : NaturalCupNodes (liftState vacuumTwo) (liftState vacuumZero) 0 := by |
| intro X i hi hX |
| have : i=0 := by omega |
| subst i |
| rw [naturalPrescription_last_cup] |
| exact vacuumTwo_firstCup X hX |
|
|
| theorem vacuumThree_nodes : NaturalCupNodes (liftState vacuumThree) (liftState vacuumOne) 1 := by |
| intro X i hi hX |
| interval_cases i |
| · funext s |
| simpa [naturalPrescription,leftFusion,braidTrain,cupLinear,siteR,pairLinear,pairAction, |
| replacePair,skipTwo,skipOne,hX,update_nat_decidable,mul_assoc] using baseThree_node (X 0) (X 1) (s 0) (s 1) (s 2) |
| · rw [naturalPrescription_last_cup] |
| exact vacuumThree_lastCup X hX |
|
|
| theorem vacuumOne_lastZero : LastZeroReduction (liftState vacuumOne) (liftState vacuumZero) 0 := by |
| intro X s hX hs |
| simp [prefixProduct] |
| |
| theorem vacuumTwo_lastZero : LastZeroReduction (liftState vacuumTwo) (liftState vacuumOne) 1 := by |
| intro X s hX hs |
| cases h0 : s 0 < |
| simp_all [baseTwo,prefixProduct] |
| |
| theorem baseThree_lastZero (x y : ℂ) (a b c : Spin) (hc : c≠Spin.plus) : |
| baseThree x y 0 a b c=(if c=Spin.zero then x*y*baseTwo x y a b else 0) := by |
| cases a < |
| simp_all [baseThree,baseThreeFactor,baseTwo] |
| all_goals vacuum_ring_finish |
|
|
| theorem vacuumThree_lastZero : LastZeroReduction (liftState vacuumThree) (liftState vacuumTwo) 2 := by |
| intro X s hX hs |
| simpa [prefixProduct,hX,Finset.prod_range_succ] using baseThree_lastZero (X 0) (X 1) (s 0) (s 1) (s 2) hs |
|
|
| end PolynomialVacuumAlgebra |
|
|
| end OAI |
|
|
