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a72760f | 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 | import OAI.NumberTheory.Ostmann.Tree.CycleQuartetActionsSelected
namespace OAI
noncomputable section
namespace Ostmann.Tree.CycleQuartet
def position (a b : Bool) : Leaves 2 := Fin.cons a (Fin.cons b Fin.elim0)
@[simp] theorem position_zero (a b : Bool) : position a b 0=a := rfl
@[simp] theorem position_one (a b : Bool) : position a b 1=b := rfl
@[simp] theorem position_eq_iff (a b x y : Bool) : position a b=position x y ↔ a=x ∧ b=y := by
constructor
· intro h
exact ⟨congrFun h 0,congrFun h 1⟩
· rintro ⟨rfl,rfl⟩
rfl
theorem position_eta (w : Leaves 2) : position (w 0) (w 1)=w := by
funext i
fin_cases i <;> rfl
theorem exists_position (w : Leaves 2) : ∃a b,w=position a b :=
⟨w 0,w 1,(position_eta w).symm⟩
variable {F : Type*} [Field F]
theorem twoLeafAct_swap (u v : Leaves 2) (z : Fˣ) (M : Leaves 2 → Fˣ) :
twoLeafAct v u z M=twoLeafAct u v z⁻¹ M := by
funext w
simp only [twoLeafAct,inv_inv]
rw [mul_comm (if w=v then z else 1) (if w=u then z⁻¹ else 1)]
theorem twoLeafAct_cross (a b : Bool) (z : Fˣ) (M : Leaves 2 → Fˣ) :
twoLeafAct (position false a) (position true b) z M=Quartet.moveCross M a b z := by
funext w
obtain ⟨x,y,rfl⟩ := exists_position w
cases a <;> cases b <;> cases x <;> cases y <;>
simp [twoLeafAct,Quartet.moveCross,Quartet.scalePair,Quartet.scaleLeaf,
Density.join,Density.left,Density.right,position,mul_comm]
theorem twoLeafAct_same (a : Bool) (z : Fˣ) (M : Leaves 2 → Fˣ) :
twoLeafAct (position a false) (position a true) z M=Quartet.NodeInput.moveSame M a z := by
funext w
obtain ⟨x,y,rfl⟩ := exists_position w
cases a <;> cases x <;> cases y <;>
simp [twoLeafAct,Quartet.NodeInput.moveSame,Quartet.movePair,Quartet.scaleLeaf,
Density.join,Density.left,Density.right,position,mul_comm]
inductive Kind where
| same (side inverse : Bool)
| cross (leftLeaf rightLeaf inverse : Bool)
deriving DecidableEq
def Kind.act (kind : Kind) (z : Fˣ) (M : Leaves 2 → Fˣ) : Leaves 2 → Fˣ :=
match kind with
| .same side inverse => Quartet.NodeInput.moveSame M side (if inverse then z⁻¹ else z)
| .cross a b inverse => Quartet.moveCross M a b (if inverse then z⁻¹ else z)
def classifyPositions (u v : Leaves 2) : Kind :=
match u 0,v 0 with
| false,false => .same false (u 1)
| true,true => .same true (u 1)
| false,true => .cross (u 1) (v 1) false
| true,false => .cross (v 1) (u 1) true
theorem twoLeafAct_classify (u v : Leaves 2) (huv : u≠v) (z : Fˣ) (M : Leaves 2 → Fˣ) :
twoLeafAct u v z M=(classifyPositions u v).act z M := by
obtain ⟨a,b,rfl⟩ := exists_position u
obtain ⟨x,y,rfl⟩ := exists_position v
cases a <;> cases b <;> cases x <;> cases y
all_goals try exact False.elim (huv rfl)
all_goals simp only [classifyPositions,position_zero,position_one,Kind.act,
Bool.false_eq_true,ite_false,ite_true]
all_goals first
| exact twoLeafAct_same _ _ _
| exact twoLeafAct_cross _ _ _ _
| rw [twoLeafAct_swap,twoLeafAct_same]
| rw [twoLeafAct_swap,twoLeafAct_cross]
variable {k b : ℕ} (P : LeafPartition (k+2) b)
(c : BalancedSelection P.label (quartetCut k).label)
def kind (v : cycleQuartets P c) : Kind :=
classifyPositions (positivePosition P c v) (negativePosition P c v)
theorem localAct_eq_kind (v : cycleQuartets P c) (z : Fˣ) (M : Leaves 2 → Fˣ) :
localAct P c v.val z M=(kind P c v).act z M := by
rw [localAct_eq_twoLeafAct]
exact twoLeafAct_classify _ _ (positions_distinct P c v) z M
end Ostmann.Tree.CycleQuartet
end
end OAI
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