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8e4d533 | 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 | 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
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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