| import Mathlib |
|
|
| namespace OAI |
|
|
| section |
| section |
| section |
| noncomputable section |
| namespace CannonTriangleBoundary |
| open Finset |
|
|
| def cycle {V R : Type*} [Add R] (f : V → V → R) (a b c : V) := f a b+f b c+f c a |
|
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| def signed {R : Type*} [Neg R] (b : Bool) (x : R) := if b then x else -x |
|
|
| lemma triangle_boundary_signs {D V : Type*} [Fintype D] |
| (a b c : D → V) {u v w : V} (huv : u≠v) (huw : u≠w) (hvw : v≠w) |
| (s t r : Bool) |
| (h : ∀ f : V → V → ℤ,(∀ x y,f y x= -f x y) → |
| (∑ d,cycle f (a d) (b d) (c d))=signed s (f u v)+signed t (f v w)+signed r (f w u)) : |
| s=t ∧ s=r := by |
| classical |
| let fu : V → V → ℤ := fun x y=>(if x=u then 1 else 0)-(if y=u then 1 else 0) |
| let fv : V → V → ℤ := fun x y=>(if x=v then 1 else 0)-(if y=v then 1 else 0) |
| have H1 := h fu (by intro x y; dsimp [fu]; ring) |
| have H2 := h fv (by intro x y; dsimp [fv]; ring) |
| have hzero (g : V → ℤ) (x y z : V) : cycle (fun p q=>g p-g q) x y z=0 := by unfold cycle; ring |
| simp only [fu,hzero,Finset.sum_const_zero] at H1 |
| simp only [fv,hzero,Finset.sum_const_zero] at H2 |
| cases s <;> cases t <;> cases r <;> |
| simp [signed,huv,Ne.symm huv,Ne.symm huw,Ne.symm hvw] at H1 H2 |
| all_goals exact ⟨rfl,rfl⟩ |
|
|
| end CannonTriangleBoundary |
|
|
| end |
| end |
| end |
| end |
|
|
| end OAI |
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|
