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97c3405 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 | 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
end
end OAI
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