| import Mathlib |
| import OAI.Geometry.Cannon.PlanarGraphs.Next |
|
|
| namespace OAI |
|
|
| section |
| section |
| section |
| noncomputable section |
| namespace CannonCycleOrder |
|
|
| theorem eq_of_steps {n : ℕ} (hn : 0<n) {X : Type*} (z : Fin n → X) |
| (hstep : ∀ i,z i=z (next hn i)) (i j : Fin n) : z i=z j := by |
| have hc (k : ℕ) (hk : k<n) : z ⟨k,hk⟩=z ⟨0,hn⟩ := by |
| induction k with |
| | zero => rfl |
| | succ k ih => |
| have hnx : next hn ⟨k,by omega⟩=(⟨k+1,hk⟩ : Fin n) := by |
| simp only [next,dite_eq_left hk] |
| have he : z (⟨k+1,hk⟩ : Fin n)=z ⟨k,by omega⟩ := by |
| rw [←hnx] |
| exact (hstep ⟨k,by omega⟩).symm |
| exact he.trans (ih (by omega)) |
| exact (hc i.val i.isLt).trans (hc j.val j.isLt).symm |
|
|
| theorem eq_of_steps_except {n : ℕ} (hn : 0 < n) {X : Type*} (z : Fin n → X) |
| (o : Fin n) (hstep : ∀ i,i≠o → next hn i≠o → z i=z (next hn i)) |
| {i j : Fin n} (hi : i≠o) (hj : j≠o) : z i=z j := by |
| let rot (k : ℕ) : Fin n := ⟨(o.val+1+k)%n,Nat.mod_lt _ hn⟩ |
| have hrnext (k : ℕ) : rot (k+1)=next hn (rot k) := by |
| apply Fin.ext |
| rw [next_mod] |
| dsimp [rot] |
| rw [←Nat.add_assoc (o.val+1) k 1, Nat.add_mod (o.val+1+k) 1, |
| Nat.add_mod ((o.val+1+k)%n) 1, Nat.mod_mod] |
| have hrne (k : ℕ) (hk : k < n-1) : rot k≠o := by |
| intro hh |
| have he := congrArg Fin.val hh |
| change (o.val+1+k)%n=o.val at he |
| by_cases h : o.val+1+k < n |
| · rw [Nat.mod_eq_of_lt h] at he; omega |
| · have hb : o.val+1+k < 2*n := by omega |
| have hm : (o.val+1+k)%n=o.val+1+k-n := Nat.mod_eq_sub_mod (by omega) |>.trans (Nat.mod_eq_of_lt (by omega)) |
| rw [hm] at he |
| omega |
| have hcover (i : Fin n) (hi : i≠o) : ∃ k,k < n-1 ∧ rot k=i := by |
| have hne : i.val≠o.val := fun h=>hi (Fin.ext h) |
| by_cases h : o.val < i.val |
| · refine ⟨i.val-o.val-1,by omega,?_⟩ |
| apply Fin.ext |
| change (o.val+1+(i.val-o.val-1))%n=i.val |
| rw [show o.val+1+(i.val-o.val-1)=i.val by omega,Nat.mod_eq_of_lt i.isLt] |
| · refine ⟨n-o.val-1+i.val,by omega,?_⟩ |
| apply Fin.ext |
| change (o.val+1+(n-o.val-1+i.val))%n=i.val |
| rw [show o.val+1+(n-o.val-1+i.val)=n+i.val by omega,Nat.add_mod] |
| simp [Nat.mod_eq_of_lt i.isLt] |
| have hconst (k : ℕ) (hk : k < n-1) : z (rot k)=z (rot 0) := by |
| induction k with |
| | zero => rfl |
| | succ k ih => |
| rw [hrnext] |
| exact (hstep (rot k) (hrne k (by omega)) (by rw [←hrnext]; exact hrne (k+1) hk)).symm.trans (ih (by omega)) |
| obtain ⟨a,ha,hra⟩ := hcover i hi |
| obtain ⟨b,hb,hrb⟩ := hcover j hj |
| rw [←hra,←hrb] |
| exact (hconst a ha).trans (hconst b hb).symm |
|
|
| end CannonCycleOrder |
|
|
| end |
| end |
| end |
| end |
|
|
| end OAI |
|
|
