| import Mathlib |
|
|
| namespace OAI |
|
|
| section |
| section |
| section |
| noncomputable section |
| namespace CannonPackingGeometry |
| open Set Metric |
|
|
| lemma half_cos_pos {a : ℝ} (ha : a∈Ioo (0:ℝ) Real.pi) : 0<Real.cos (a/2) := |
| Real.cos_pos_of_mem_Ioo ⟨by linarith [Real.pi_pos,ha.1],by linarith [ha.2]⟩ |
|
|
| theorem intrinsic_inradius_identity {a b c x y z q : ℝ} |
| (ha : a∈Ioo (0:ℝ) Real.pi) (hb : b∈Ioo (0:ℝ) Real.pi) |
| (hc : c∈Ioo (0:ℝ) Real.pi) (hsum : a+b+c=Real.pi) |
| (hx : 0<x) (hy : 0<y) (hz : 0<z) (_hq : 0<q) |
| (hax : Real.tan (a/2)=q/x) (hby : Real.tan (b/2)=q/y) |
| (hcz : Real.tan (c/2)=q/z) : q^2*(x+y+z)=x*y*z := by |
| have hca := half_cos_pos ha |
| have hcb := half_cos_pos hb |
| have hcc := half_cos_pos hc |
| have hsa : Real.sin (a/2)=q/x*Real.cos (a/2) := by |
| rw [Real.tan_eq_sin_div_cos] at hax |
| exact (div_eq_iff hca.ne').mp hax |
| have hsb : Real.sin (b/2)=q/y*Real.cos (b/2) := by |
| rw [Real.tan_eq_sin_div_cos] at hby |
| exact (div_eq_iff hcb.ne').mp hby |
| have hsc : Real.sin (c/2)=q/z*Real.cos (c/2) := by |
| rw [Real.tan_eq_sin_div_cos] at hcz |
| exact (div_eq_iff hcc.ne').mp hcz |
| have hangle : a/2+b/2+c/2=Real.pi/2 := by linarith |
| have hzero : Real.cos (a/2+b/2+c/2)=0 := by rw [hangle,Real.cos_pi_div_two] |
| rw [Real.cos_add,Real.cos_add,Real.sin_add,hsa,hsb,hsc] at hzero |
| have hprod : Real.cos (a/2)*Real.cos (b/2)*Real.cos (c/2)≠0 := by positivity |
| have hid : (1-q/y*(q/z)-q/x*(q/z)-q/x*(q/y))* |
| (Real.cos (a/2)*Real.cos (b/2)*Real.cos (c/2))=0 := by nlinarith only [hzero] |
| have he := (mul_eq_zero.mp hid).resolve_right hprod |
| field_simp [hx.ne',hy.ne',hz.ne'] at he |
| nlinarith only [he] |
|
|
| theorem intrinsic_cosine_law {a x y z q : ℝ} |
| (ha : a∈Ioo (0:ℝ) Real.pi) (hx : 0<x) (_hy : 0<y) (_hz : 0<z) |
| (hax : Real.tan (a/2)=q/x) (hid : q^2*(x+y+z)=x*y*z) : |
| 2*(x+y)*(x+z)*Real.cos a=(x+y)^2+(x+z)^2-(y+z)^2 := by |
| have hca := half_cos_pos ha |
| have hsa : Real.sin (a/2)=q/x*Real.cos (a/2) := by |
| rw [Real.tan_eq_sin_div_cos] at hax |
| exact (div_eq_iff hca.ne').mp hax |
| have ht := Real.sin_sq_add_cos_sq (a/2) |
| rw [hsa] at ht |
| have ht' : (q^2+x^2)*Real.cos (a/2)^2=x^2 := by |
| field_simp [hx.ne'] at ht |
| nlinarith only [ht] |
| have hc : Real.cos a=2*Real.cos (a/2)^2-1 := by |
| convert Real.cos_two_mul (a/2) using 1 |
| congr 1 |
| ring |
| rw [hc] |
| have hdiv : (x+y)*(x+z)=(x+y+z)*(q^2+x^2)/x := by |
| apply (eq_div_iff hx.ne').mpr |
| nlinarith only [hid] |
| rw [show 2*(x+y)*(x+z)=2*((x+y)*(x+z)) by ring,hdiv] |
| field_simp [hx.ne'] |
| linear_combination 4*(x+y+z)*ht'-2*hid |
|
|
| def trianglePoint (x z a : ℝ) : ℂ := ⟨(x+z)*Real.cos a,(x+z)*Real.sin a⟩ |
|
|
| theorem intrinsic_triangle {a b c x y z q : ℝ} |
| (ha : a∈Ioo (0:ℝ) Real.pi) (hb : b∈Ioo (0:ℝ) Real.pi) |
| (hc : c∈Ioo (0:ℝ) Real.pi) (hsum : a+b+c=Real.pi) |
| (hx : 0<x) (hy : 0<y) (hz : 0<z) (hq : 0<q) |
| (hax : Real.tan (a/2)=q/x) (hby : Real.tan (b/2)=q/y) |
| (hcz : Real.tan (c/2)=q/z) : |
| dist (0:ℂ) (x+y : ℂ)=x+y ∧ |
| dist (0:ℂ) (trianglePoint x z a)=x+z ∧ |
| dist (x+y : ℂ) (trianglePoint x z a)=y+z ∧ |
| 0<(trianglePoint x z a).im := by |
| have hid := intrinsic_inradius_identity ha hb hc hsum hx hy hz hq hax hby hcz |
| have hcos := intrinsic_cosine_law ha hx hy hz hax hid |
| have hsincos := Real.sin_sq_add_cos_sq a |
| have hnorm : ‖trianglePoint x z a‖^2=(x+z)^2 := by |
| rw [Complex.sq_norm] |
| simp only [trianglePoint,Complex.normSq_apply] |
| nlinarith only [hsincos] |
| have hnorm' : ‖(x+y : ℂ)-trianglePoint x z a‖^2=(y+z)^2 := by |
| rw [Complex.sq_norm] |
| simp only [trianglePoint,Complex.normSq_apply,Complex.sub_re,Complex.sub_im, |
| Complex.add_re,Complex.add_im,Complex.ofReal_re,Complex.ofReal_im] |
| nlinarith only [hsincos,hcos] |
| refine ⟨?_,?_,?_,?_⟩ |
| · rw [dist_zero_left,←Complex.ofReal_add,Complex.norm_real,Real.norm_eq_abs,abs_of_pos (add_pos hx hy)] |
| · rw [dist_zero_left] |
| nlinarith [norm_nonneg (trianglePoint x z a)] |
| · rw [dist_eq_norm] |
| nlinarith [norm_nonneg ((x+y : ℂ)-trianglePoint x z a)] |
| · exact mul_pos (add_pos hx hz) (Real.sin_pos_of_pos_of_lt_pi ha.1 ha.2) |
| end CannonPackingGeometry |
| |
| end |
| end |
| section |
| noncomputable section |
| namespace CannonPackingGeometry |
| open Set |
| |
| def cornerAngle {V A : Type*} (vertex : A × Fin 3 → V) (θ : A × Fin 3 → ℝ) |
| (f : A) (v : V) : ℝ := by |
| classical |
| exact ∑ k : Fin 3, if vertex (f,k)=v then θ (f,k) else 0 |
| |
| lemma cornerAngle_at {V A : Type*} (vertex : A × Fin 3 → V) (θ : A × Fin 3 → ℝ) |
| (hinj : ∀ f,Function.Injective (fun k=>vertex (f,k))) (f : A) (k : Fin 3) : |
| cornerAngle vertex θ f (vertex (f,k))=θ (f,k) := by |
| classical |
| unfold cornerAngle |
| rw [Finset.sum_eq_single k] |
| · simp |
| · intro j hj hjk |
| simp only [ite_eq_right (fun h=>hjk (hinj f h))] |
| · simp |
| |
| lemma cornerAngle_zero {V A : Type*} (vertex : A × Fin 3 → V) (θ : A × Fin 3 → ℝ) |
| (f : A) (v : V) (h : ¬∃ k,vertex (f,k)=v) : cornerAngle vertex θ f v=0 := by |
| classical |
| apply Finset.sum_eq_zero |
| intro k hk |
| exact ite_eq_right (fun hk=>h ⟨k,hk⟩) |
| |
| lemma sum_cornerAngle {V A : Type*} [Fintype A] [DecidableEq V] (vertex : A × Fin 3 → V) |
| (θ : A × Fin 3 → ℝ) (v : V) : |
| (∑ f,cornerAngle vertex θ f v)=∑ i : A × Fin 3,if vertex i=v then θ i else 0 := by |
| classical |
| simp only [cornerAngle,Fintype.sum_prod_type] |
| apply Finset.sum_congr rfl |
| intro f hf |
| apply Finset.sum_congr rfl |
| intro k hk |
| by_cases h : vertex (f,k)=v <;> simp [h] |
| |
| theorem intrinsic_cosine_law_fin3 (θ r : Fin 3 → ℝ) (q : ℝ) |
| (hθ : ∀ k,θ k∈Ioo (0:ℝ) Real.pi) (hs : ∑ k,θ k=Real.pi) |
| (hr : ∀ k,0<r k) (hq : 0<q) |
| (ht : ∀ k,Real.tan (θ k/2)=q/r k) {i j k : Fin 3} |
| (hij : i≠j) (hik : i≠k) (hjk : j≠k) : |
| 2*(r i+r j)*(r i+r k)*Real.cos (θ i)= |
| (r i+r j)^2+(r i+r k)^2-(r j+r k)^2 := by |
| have hs' : θ 0+θ 1+θ 2=Real.pi := by simpa only [Fin.sum_univ_three] using hs |
| have hid := intrinsic_inradius_identity (hθ 0) (hθ 1) (hθ 2) hs' |
| (hr 0) (hr 1) (hr 2) hq (ht 0) (ht 1) (ht 2) |
| apply intrinsic_cosine_law (hθ i) (hr i) (hr j) (hr k) (ht i) |
| fin_cases i <;> fin_cases j <;> fin_cases k <;> simp_all |
| all_goals nlinarith only [hid] |
| end CannonPackingGeometry |
| |
| end |
| end |
| end |
| end |
| |
| end OAI |
| |
