| import Mathlib |
| import OAI.Geometry.Cannon.PlanarGraphs.Next |
|
|
| namespace OAI |
|
|
| section |
| section |
| section |
| noncomputable section |
| namespace CannonCycleOrder |
|
|
| def cutOrder {n : ℕ} (hn : 0<n) (o : Fin n) (k : ℕ) : Fin n := |
| ⟨(o.val+1+k)%n,Nat.mod_lt _ hn⟩ |
|
|
| lemma cutOrder_next {n : ℕ} (hn : 0<n) (o : Fin n) (k : ℕ) : |
| cutOrder hn o (k+1)=next hn (cutOrder hn o k) := by |
| apply Fin.ext |
| rw [next_mod] |
| dsimp [cutOrder] |
| 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] |
|
|
| lemma cutOrder_last {n : ℕ} (hn : 0<n) (o : Fin n) : cutOrder hn o (n-1)=o := by |
| apply Fin.ext |
| change (o.val+1+(n-1))%n=o.val |
| rw [show o.val+1+(n-1)=n+o.val by omega,Nat.add_mod] |
| simp [Nat.mod_eq_of_lt o.isLt] |
|
|
| lemma cutOrder_first {n : ℕ} (hn : 0<n) (o : Fin n) : cutOrder hn o 0=next hn o := by |
| apply Fin.ext |
| rw [next_mod] |
| simp [cutOrder] |
|
|
| lemma cutOrder_ne {n : ℕ} (hn : 0<n) (o : Fin n) {k : ℕ} (hk : k<n-1) : |
| cutOrder hn o 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 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 |
|
|
| lemma cutOrder_inj {n : ℕ} (hn : 0<n) (o : Fin n) {k l : ℕ} (hk : k<n) (hl : l<n) |
| (he : cutOrder hn o k=cutOrder hn o l) : k=l := by |
| have hh := congrArg Fin.val he |
| change (o.val+1+k)%n=(o.val+1+l)%n at hh |
| have hm (j : ℕ) (hj : j<n) (hh : n≤o.val+1+j) : |
| (o.val+1+j)%n=o.val+1+j-n := Nat.mod_eq_sub_mod hh |>.trans (Nat.mod_eq_of_lt (by omega)) |
| by_cases h1 : o.val+1+k<n <;> by_cases h2 : o.val+1+l<n |
| · rw [Nat.mod_eq_of_lt h1,Nat.mod_eq_of_lt h2] at hh; omega |
| · rw [Nat.mod_eq_of_lt h1,hm l hl (by omega)] at hh; omega |
| · rw [hm k hk (by omega),Nat.mod_eq_of_lt h2] at hh; omega |
| · rw [hm k hk (by omega),hm l hl (by omega)] at hh; omega |
|
|
| lemma cutOrder_cover {n : ℕ} (hn : 0<n) (o i : Fin n) (hi : i≠o) : |
| ∃ k,k<n-1 ∧ cutOrder hn o 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] |
|
|
| lemma sum_cutOrder {n : ℕ} (hn : 0<n) (o : Fin n) (f : Fin n → ℝ) (ho : f o=0) : |
| ∑ k∈Finset.range (n-1),f (cutOrder hn o k)=∑ i,f i := by |
| classical |
| have he : (∑ k∈Finset.range (n-1),f (cutOrder hn o k))=∑ i∈Finset.univ.erase o,f i := by |
| apply Finset.sum_bij (fun k _=>cutOrder hn o k) |
| · intro k hk; exact Finset.mem_erase.mpr ⟨cutOrder_ne hn o (Finset.mem_range.mp hk),Finset.mem_univ _⟩ |
| · intro k hk l hl he |
| exact cutOrder_inj hn o (by have := Finset.mem_range.mp hk; omega) (by have := Finset.mem_range.mp hl; omega) he |
| · intro i hi |
| obtain ⟨k,hk,he⟩:=cutOrder_cover hn o i (Finset.mem_erase.mp hi).1 |
| exact ⟨k,Finset.mem_range.mpr hk,he⟩ |
| · intro k hk; rfl |
| rw [he] |
| have hh:=Finset.sum_erase_add Finset.univ f (Finset.mem_univ o) |
| rw [ho,add_zero] at hh |
| exact hh |
|
|
| end CannonCycleOrder |
|
|
| end |
| end |
| end |
| end |
|
|
| end OAI |
|
|
