| import Mathlib | |
| import OAI.Combinatorics.Chromatic.Shuffle.PrimitiveRootClosure | |
| import OAI.Combinatorics.Chromatic.Shuffle.InputProduct | |
| namespace OAI | |
| section | |
| namespace ElementaryPositivity.RawShuffle | |
| open SlopeArithmetic QuantumTorus WeightedTorusSeries WallUnits PowerSeries | |
| noncomputable section | |
| variable {I J M : Type*} [Fintype I] [DecidableEq I] [Fintype J] [AddCommGroup M] | |
| variable (Ω : M→+M→+ℤ) (hΩ : ∀p,Ω p p=0) | |
| variable (p : I→M) (κ : I→ℤ) (ε : I→Bool) | |
| variable (w : I→ℕ) [Fact (∀i,0<w i)] | |
| variable (c η : I→ℝ) (hc : ∀i,0<c i) (t : ℝ) | |
| variable (hp : ∀i j,(Ω (p i) (p j):ℝ)=t*(η i*c j-c i*η j)) | |
| variable (l : List I) (hn : l.Nodup) (hall : l.toFinset=Finset.univ) | |
| variable (hl : l.Pairwise (fun i j=>0≤Ω (p i) (p j))) | |
| variable (C : (J→ℤ)→+M) (hroot : ∀d,HasRootDegree C (weight w d) (rootSum p d)) | |
| variable (h : M→+ℝ) (θ : ℝ) (hh : ∀d,h (rootSum p d)=slopeValue c η θ d) | |
| variable (V : M→Prop) (hv : ∀d∈slopeDimensions c η hc θ,V (rootSum p d)) | |
| include hΩ hc hp hn hall hl hroot hh hv in | |
| theorem geometric_zero_rootClosure : | |
| InPrecisionClosure LaurentRay.vUnit Ω (literalRootProducts Ω C V) | |
| (PowerSeriesSplit.zeroFactor (positiveProject LaurentRay.vUnit Ω h) | |
| (zeroProject LaurentRay.vUnit Ω h) | |
| ((l.map (literalAxis κ ε Ω (rootSum p) w)).prod)) := by | |
| let a:=geometricQuiver Ω p ε | |
| let : Fact (∀θ,SlopeEulerSymmetric a c η θ) := | |
| ⟨geometricQuiver_symmetric Ω p ε hΩ c η hc t hp⟩ | |
| have H:=literalInput_ordered a κ ε Ω (rootSum p) w | |
| (geometricQuiver_euler Ω p ε hΩ) l hn hall | |
| (geometricQuiver_pairwise Ω p ε hΩ l hn hl) | |
| rw [←H] | |
| exact finiteInput_zero_rootClosure a κ c η hc w Ω (rootSum p) C | |
| (geometricQuiver_euler Ω p ε hΩ) hroot θ V hv h hh ε | |
| (geometricQuiver_diag Ω p ε) | |
| end | |
| end ElementaryPositivity.RawShuffle | |
| end | |
| section | |
| namespace ElementaryPositivity.RawShuffle | |
| open SlopeArithmetic QuantumTorus WeightedTorusSeries WallUnits | |
| noncomputable section | |
| variable {I J M : Type*} [Fintype I] [DecidableEq I] [Fintype J] [AddCommGroup M] | |
| variable (p : I→M) (w : I→ℕ) (C : (J→ℤ)→+M) | |
| lemma rootDegree_finset_sum {A : Type*} (s : Finset A) (n : A→ℕ) (m : A→M) | |
| (h : ∀i∈s,HasRootDegree C (n i) (m i)) : | |
| HasRootDegree C (∑i∈s,n i) (∑i∈s,m i) := by | |
| classical | |
| induction s using Finset.induction_on with | |
| | empty=>simpa using rootDegree_zero C | |
| | @insert a s ha ih=> | |
| simp only [Finset.sum_insert ha] | |
| exact rootDegree_add C (h a (by simp)) (ih (fun i hi=>h i (by simp [hi]))) | |
| omit [DecidableEq I] in | |
| lemma rootSum_rootDegree (hp : ∀i,HasRootDegree C (w i) (p i)) (d : I→ℕ) : | |
| HasRootDegree C (weight w d) (rootSum p d) := by | |
| exact rootDegree_finset_sum C Finset.univ (fun i=>d i*w i) (fun i=>d i • p i) | |
| (fun i _=>rootDegree_nsmul C (hp i) (d i)) | |
| omit [DecidableEq I] in | |
| lemma rootSum_mem (V : AddSubmonoid M) (hp : ∀i,p i∈V) (d : I→ℕ) : rootSum p d∈V := by | |
| exact V.sum_mem (fun i _=>V.nsmul_mem (hp i) (d i)) | |
| omit [DecidableEq I] in | |
| lemma rootSum_slope_zero (h : M→+ℝ) (c : I→ℝ) (d : I→ℕ) : | |
| h (rootSum p d)=slopeValue c (fun i=>h (p i)) 0 d := by | |
| simp only [rootSum,AddMonoidHom.coe_mk,ZeroHom.coe_mk,map_sum,map_nsmul, | |
| slopeValue,zero_mul,sub_zero,mass,nsmul_eq_mul] | |
| end | |
| end ElementaryPositivity.RawShuffle | |
| end | |
| end OAI | |