| import Mathlib |
| import OAI.Combinatorics.Chromatic.Walls.LaurentGradedFibers |
| import OAI.Combinatorics.Chromatic.Witness.GraphDecoratedHistories |
| import OAI.Combinatorics.Chromatic.Histories.AnchorEmptyMask |
|
|
| namespace OAI |
|
|
| section |
| namespace ElementaryPositivity |
| open scoped BigOperators |
| open Classical |
| noncomputable section |
| variable {A:Type*} [Fintype A] {n:ℕ} |
|
|
| def profilePartition (μ:A →₀ ℕ) (hμ:μ.degree=n):Nat.Partition n:= |
| Nat.Partition.ofSums n (Finset.univ.val.map μ) (by |
| change (∑i:A,μ i)=n |
| rw [←Finsupp.degree_eq_sum] |
| exact hμ) |
|
|
| lemma esymmPart_profile {B R:Type*} [Fintype B] [CommSemiring R] |
| (μ:A →₀ ℕ) (hμ:μ.degree=n) : |
| MvPolynomial.esymmPart B R (profilePartition μ hμ)=∏i:A,MvPolynomial.esymm B R (μ i) := by |
| have H:∀l:Multiset ℕ,((l.filter (·≠0)).map (MvPolynomial.esymm B R)).prod= |
| (l.map (MvPolynomial.esymm B R)).prod:=by |
| intro l |
| induction l using Multiset.induction_on with |
| | empty=>simp |
| | cons k l ih=> |
| by_cases hk:k=0 |
| · subst k; simp [MvPolynomial.esymm_zero,ih] |
| · simp [hk,ih] |
| unfold MvPolynomial.esymmPart profilePartition |
| rw [Nat.Partition.ofSums_parts,H,Multiset.map_map] |
| rfl |
| end |
| end ElementaryPositivity |
|
|
| end |
| section |
| namespace ElementaryPositivity.ExactMasks |
| open Classical |
| noncomputable section |
| instance requiredFintype {A I:Type*} [Fintype A] [DecidableEq I] (mask:A → Finset I) (J:Finset I) : |
| Fintype (Required mask J):=inferInstanceAs (Fintype {a:A // J⊆mask a}) |
| instance exactFintype {A I:Type*} [Fintype A] [DecidableEq I] (mask:A → Finset I) (J:Finset I) : |
| Fintype (Exact mask J):=inferInstanceAs (Fintype {a:A // mask a=J}) |
| end |
| end ElementaryPositivity.ExactMasks |
| namespace ElementaryPositivity.TriangularDynamics |
| open Packets |
| open scoped BigOperators |
| open Classical |
| noncomputable section |
| variable {n:ℕ} (G:NaturalUnitIntervalGraph n) |
| |
| def requiredHistoryMatching (J:Finset (Fin (n-1))) : |
| ExactMasks.GradedEquiv |
| (fun w:ExactMasks.Required G.wordMask J=>G.graphInversions w.val) |
| (fun w:ExactMasks.Required (decoratedMask G) J=>decoratedEnergy G w.val) := |
| LaurentPositive.gradedEquivOfEnergySum G.edgeCount _ _ (required_energy_sum G J) |
| |
| def exactHistoryMatching (J:Finset (Fin (n-1))) : |
| ExactMasks.GradedEquiv |
| (fun w:ExactMasks.Exact G.wordMask J=>G.graphInversions w.val) |
| (fun w:ExactMasks.Exact (decoratedMask G) J=>decoratedEnergy G w.val) := by |
| letI:Encodable (Equiv.Perm (Fin n)):=Fintype.toEncodable _ |
| exact ExactMasks.refine G.wordMask (decoratedMask G) G.graphInversions (decoratedEnergy G) |
| (requiredHistoryMatching G) J |
| |
| def admissibleExactEquiv : {w:Equiv.Perm (Fin n) // G.Admissible w} ≃ ExactMasks.Exact G.wordMask ∅ where |
| toFun w:=⟨w.val,(G.wordMask_empty w.val).mpr w.property⟩ |
| invFun w:=⟨w.val,(G.wordMask_empty w.val).mp w.property⟩ |
| left_inv _:=rfl |
| right_inv _:=rfl |
|
|
| def emptyDecoratedEquiv : ExactMasks.Exact (decoratedMask G) ∅ ≃ DecoratedLeaf G where |
| toFun w:=w.val.1 |
| invFun d:=⟨⟨d,((emptyAnchorEquiv.{0,1} (historyProfile G d.1) (historyProfile_degree G d.1)).symm PUnit.unit.{1}).val⟩, |
| ((emptyAnchorEquiv.{0,1} (historyProfile G d.1) (historyProfile_degree G d.1)).symm PUnit.unit.{1}).property⟩ |
| left_inv w:=by |
| rcases w with ⟨⟨d,w⟩,hw⟩ |
| have H: (emptyAnchorEquiv.{0,1} (historyProfile G d.1) (historyProfile_degree G d.1)).symm PUnit.unit.{1}= |
| (⟨w,hw⟩:EmptyAnchor (n:=n) (historyProfile G d.1)):= |
| (emptyAnchorEquiv.{0,1} (historyProfile G d.1) (historyProfile_degree G d.1)).symm_apply_apply ⟨w,hw⟩ |
| apply Subtype.ext |
| exact congrArg (fun z:EmptyAnchor (n:=n) (historyProfile G d.1)=> (⟨d,z.val⟩:DecoratedWord G)) H |
| right_inv _:=rfl |
|
|
| def admissibleHistoryEquiv : {w:Equiv.Perm (Fin n) // G.Admissible w} ≃ DecoratedLeaf G:= |
| (admissibleExactEquiv G).trans ((exactHistoryMatching G ∅).val.trans (emptyDecoratedEquiv G)) |
|
|
| lemma admissibleHistoryEquiv_energy (w:{w:Equiv.Perm (Fin n) // G.Admissible w}) : |
| leafEnergy G (admissibleHistoryEquiv G w)=G.graphInversions w.val:= |
| (exactHistoryMatching G ∅).property (admissibleExactEquiv G w) |
|
|
| def leafPartition (d:DecoratedLeaf G):Nat.Partition n:= |
| profilePartition (historyProfile G d.1) (historyProfile_degree G d.1) |
|
|
| def admissiblePartition (w:{w:Equiv.Perm (Fin n) // G.Admissible w}):Nat.Partition n:= |
| leafPartition G (admissibleHistoryEquiv G w) |
| end |
| end ElementaryPositivity.TriangularDynamics |
|
|
| end |
|
|
| end OAI |
|
|
