| import Mathlib |
| import OAI.Combinatorics.Chromatic.QuantumTorus.Admissible |
| import OAI.Combinatorics.Chromatic.Histories.SortedFiberPerm |
|
|
| namespace OAI |
|
|
| section |
| namespace ElementaryPositivity.NaturalUnitIntervalGraph |
| open Packets |
| open scoped BigOperators |
| open Classical |
| noncomputable section |
| variable {n:ℕ} (G:NaturalUnitIntervalGraph n) |
|
|
| def nonedgeLT (a b:Fin n):Prop:=a<b ∧ ¬G.Edge a b |
| lemma nonedgeLT_trans : ∀ ⦃first middle last : Fin n⦄, |
| G.nonedgeLT first middle → G.nonedgeLT middle last → G.nonedgeLT first last := by |
| intro a b c hab hbc |
| refine ⟨hab.1.trans hbc.1,?_⟩ |
| intro hac |
| rcases hac with hac|hca |
| · exact hab.2 (Or.inl ⟨hab.1,hbc.1.le.trans hac.2⟩) |
| · exact (not_lt_of_ge (hab.1.trans hbc.1).le) hca.1 |
|
|
| def wordMask (w:Equiv.Perm (Fin n)):=ascentMask G.nonedgeLT w |
| lemma wordMask_empty (w:Equiv.Perm (Fin n)) : G.wordMask w=∅ ↔ G.Admissible w := by |
| constructor |
| · intro he i j hij hlt |
| by_contra hn |
| let t:Fin (n-1):=⟨i.val,by omega⟩ |
| have hi:left t=i:=Fin.ext rfl |
| have hj:right t=j:=Fin.ext hij |
| have ht:t∈G.wordMask w:=(mem_ascentMask _ _ _).mpr (by change _ < _ ∧ ¬_; simpa only [hi,hj] using And.intro hlt hn) |
| rw [he] at ht |
| exact Finset.notMem_empty t ht |
| · intro hw |
| apply Finset.eq_empty_iff_forall_notMem.mpr |
| intro t ht |
| exact ((mem_ascentMask _ _ _).mp ht).2 |
| (hw (left t) (right t) rfl ((mem_ascentMask _ _ _).mp ht).1) |
|
|
| lemma required_sorted (J:Finset (Fin (n-1))) (w:Equiv.Perm (Fin n)) |
| (hw:J⊆G.wordMask w):Sorted (index J) w:=by |
| intro i j he hij |
| exact (inside_of_required G.nonedgeLT G.nonedgeLT_trans J w hw i j hij he).1 |
| lemma required_proper (J:Finset (Fin (n-1))) (w:Equiv.Perm (Fin n)) |
| (hw:J⊆G.wordMask w):G.Proper (fun a=>index J (w.symm a)):=by |
| intro a b hab he |
| rcases lt_trichotomy (w.symm a) (w.symm b) with h|h|h |
| · have H:=inside_of_required G.nonedgeLT G.nonedgeLT_trans J w hw _ _ h he |
| exact H.2 (by simpa only [w.apply_symm_apply] using hab) |
| · have hh:a=b:=w.symm.injective h |
| subst b |
| exact G.edge_irrefl a hab |
| · have H:=inside_of_required G.nonedgeLT G.nonedgeLT_trans J w hw _ _ h he.symm |
| exact H.2 (by simpa only [w.apply_symm_apply] using (G.edge_comm a b).mp hab) |
|
|
| lemma required_of_sorted (J:Finset (Fin (n-1))) (κ:Fin n → Fin (n+1)) |
| (hκ:G.Proper κ) (w:Equiv.Perm (Fin n)) (hw:∀i,κ (w i)=index J i) |
| (hs:Sorted (index J) w) : J⊆G.wordMask w := by |
| intro t ht |
| apply (mem_ascentMask _ _ _).mpr |
| have he:index J (left t)=index J (right t):=(index_left_right J t).mpr ht |
| refine ⟨hs _ _ he (by change t.val<t.val+1; omega),?_⟩ |
| intro hedge |
| exact hκ _ _ hedge (by rw [hw,hw,he]) |
|
|
| abbrev RequiredWord (J:Finset (Fin (n-1))):= {w:Equiv.Perm (Fin n) // J⊆G.wordMask w} |
| abbrev PacketColor (J:Finset (Fin (n-1))):= |
| {κ:Fin n → Fin (n+1) // G.Proper κ ∧ ∀c,card κ c=card (index J) c} |
|
|
| def requiredColorEquiv (J:Finset (Fin (n-1))) : G.RequiredWord J ≃ G.PacketColor J where |
| toFun w:=⟨fun a=>index J (w.val.symm a),G.required_proper J w.val w.property, |
| fun c=>(card_of_perm (index J) (fun a=>index J (w.val.symm a)) w.val (by intro i; simp) c).symm⟩ |
| invFun κ:=⟨sortedPerm (index J) κ.val (fun c=>(κ.property.2 c).symm), |
| G.required_of_sorted J κ.val κ.property.1 _ (sortedPerm_map _ _ _) (sortedPerm_sorted _ _ _)⟩ |
| left_inv w:=by |
| apply Subtype.ext |
| change sortedPerm (index J) (fun a=>index J (w.val.symm a)) _=w.val |
| apply sorted_unique (index J) (fun a=>index J (w.val.symm a)) |
| · exact sortedPerm_map (index J) (fun a=>index J (w.val.symm a)) _ |
| · intro i; simp |
| · exact sortedPerm_sorted (index J) (fun a=>index J (w.val.symm a)) _ |
| · exact G.required_sorted _ _ w.property |
| right_inv κ:=by |
| apply Subtype.ext |
| funext a |
| have H:=sortedPerm_map (index J) κ.val (fun c=>(κ.property.2 c).symm) |
| ((sortedPerm (index J) κ.val (fun c=>(κ.property.2 c).symm)).symm a) |
| simpa only [Equiv.apply_symm_apply] using H.symm |
|
|
| lemma packet_energy (J:Finset (Fin (n-1))) (κ:Fin n → Fin (n+1)) |
| (hκ:G.Proper κ) (w:Equiv.Perm (Fin n)) (hw:∀i,κ (w i)=index J i) : |
| G.coloringInversions κ=G.graphInversions w := by |
| rw [G.graphInversions_eq_card_invertedEdges,coloringInversions,invertedEdges] |
| congr 1 |
| apply Finset.filter_congr |
| intro ab hab |
| have hedge:G.Edge ab.1 ab.2:=(Finset.mem_filter.mp hab).2.2 |
| have hne:κ ab.1≠κ ab.2:=hκ _ _ hedge |
| have hcolor:∀a,κ a=index J (w.symm a):=fun a=>by simpa using hw (w.symm a) |
| rw [hcolor ab.1,hcolor ab.2] at hne ⊢ |
| constructor |
| · intro h |
| by_contra hn |
| exact (not_lt_of_ge (index_mono J (le_of_not_gt hn))) h |
| · intro h |
| exact lt_of_le_of_ne (index_mono J h.le) hne.symm |
|
|
| lemma requiredColorEquiv_energy (J:Finset (Fin (n-1))) (w:G.RequiredWord J) : |
| G.coloringInversions (G.requiredColorEquiv J w).val=G.graphInversions w.val:=by |
| apply G.packet_energy J _ (G.required_proper J w.val w.property) w.val |
| intro i |
| simp |
| end |
| end ElementaryPositivity.NaturalUnitIntervalGraph |
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| end |
|
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| end OAI |
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