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import OAI.Geometry.Cannon.Flows.State
namespace OAI
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noncomputable section
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namespace CannonMagnification
open Set Metric
lemma annular_pattern_distance {X : Type*} [MetricSpace X]
(h : X ≃ₜ X) (x y z : X) (R θ C L a d P : ℝ)
(hR : 0 < R) (hθ : 0 ≤ θ) (ha : 0 < a) (hC : 0 ≤ C)
(hexp : Expansion h x R C L)
(hy : a*R ≤ dist x y) (hz : a*R ≤ dist x z)
(hyz : dist y z ≤ d*θ*R) (hcoef : C*d ≤ P*a^2) :
dist (h y) (h z) ≤ P*θ := by
have hprod : a^2*R^2 ≤ max R (dist x y)*max R (dist x z) := by
calc
_ = (a*R)*(a*R) := by ring
_ ≤ _ := mul_le_mul (hy.trans (le_max_right _ _)) (hz.trans (le_max_right _ _))
(mul_nonneg ha.le hR.le) (hR.le.trans (le_max_left _ _))
have hh := (mul_le_mul_of_nonneg_left hprod (dist_nonneg (x := h y) (y := h z))).trans
(by simpa only [mul_assoc] using (hexp y z).1)
have hCdist := mul_le_mul_of_nonneg_left hyz (mul_nonneg hC hR.le)
have hcoeff := mul_le_mul_of_nonneg_right hcoef (mul_nonneg hθ (sq_nonneg R))
apply (mul_le_mul_iff_left₀ (mul_pos (sq_pos_of_pos ha) (sq_pos_of_pos hR))).mp
nlinarith only [hh,hCdist,hcoeff]
lemma inverse_ball_criterion {X : Type*} [MetricSpace X]
(h : X ≃ₜ X) (x y z : X) (R C L d s : ℝ)
(hR : 0 < R) (hL : 0 ≤ L) (hd : 0 < d) (hs : 0 ≤ s)
(hexp : Expansion h x R C L)
(hcoef : L*s*denominator x R y*(denominator x R y+d) < R*d)
(hyz : dist (h y) (h z) ≤ s) : dist y z < d := by
by_contra hh
have hdd : d ≤ dist y z := le_of_not_gt hh
have hD : 0 < denominator x R y := hR.trans_le (le_max_left _ _)
have hDz : 0 ≤ denominator x R z := hR.le.trans (le_max_left _ _)
have htri := denominator_triangle x y z R
have hscale : denominator x R z*d ≤ (denominator x R y+d)*dist y z := by
have hh := mul_le_mul_of_nonneg_right htri hd.le
have hh' := mul_le_mul_of_nonneg_left hdd hD.le
nlinarith only [hh,hh']
have he := (hexp y z).2
change R*dist y z ≤ L*dist (h y) (h z)*denominator x R y*denominator x R z at he
have he' := mul_le_mul_of_nonneg_right he hd.le
have hu := mul_le_mul_of_nonneg_left hyz hL
have hu' := mul_le_mul_of_nonneg_right hu (mul_nonneg hD.le (mul_nonneg hDz hd.le))
have hv := mul_le_mul_of_nonneg_left hscale (mul_nonneg (mul_nonneg hL hs) hD.le)
have hc := mul_lt_mul_of_pos_right hcoef (hd.trans_le hdd)
nlinarith only [he',hu',hv,hc]
lemma pattern_ball_containment {X : Type*} [MetricSpace X]
(h : X ≃ₜ X) (x y : X) (R θ C L B c p : ℝ)
(hR : 0 < R) (hθ : 0 < θ) (hL : 0 ≤ L) (hB : 1 ≤ B)
(hc : 0 < c) (hp : 0 ≤ p) (hsmall : c*θ ≤ 1)
(hexp : Expansion h x R C L) (hy : dist x y ≤ B*R)
(hcoef : L*p*B*(B+1) < c) :
closedBall (h y) (p*θ) ⊆ h '' ball y (c*θ*R) := by
intro z hz
refine ⟨h.symm z,?_,h.apply_symm_apply z⟩
have hD : denominator x R y ≤ B*R := max_le (by nlinarith only [hB,hR]) hy
have hDnon : 0 ≤ denominator x R y := hR.le.trans (le_max_left _ _)
have hDplus : denominator x R y+c*θ*R ≤ (B+1)*R := by
have hm := mul_le_mul_of_nonneg_right hsmall hR.le
linarith only [hD,hm]
have hco : L*(p*θ)*denominator x R y*(denominator x R y+c*θ*R) < R*(c*θ*R) := by
have hmul : L*(p*θ)*denominator x R y*(denominator x R y+c*θ*R) ≤
L*(p*θ)*(B*R)*((B+1)*R) := by
apply mul_le_mul
· exact mul_le_mul_of_nonneg_left hD (by positivity)
· exact hDplus
· positivity
· positivity
have hh := mul_lt_mul_of_pos_right hcoef (mul_pos hθ (sq_pos_of_pos hR))
nlinarith only [hmul,hh]
have hh := inverse_ball_criterion h x y (h.symm z) R C L (c*θ*R) (p*θ)
hR hL (by positivity) (by positivity) hexp hco (by
rw [h.apply_symm_apply,dist_comm]
exact mem_closedBall.mp hz)
simpa only [mem_ball,dist_comm] using hh
lemma net_pattern_ball {X : Type*} [MetricSpace X] (c z : X) (e : ℝ)
(he : 0 ≤ e) (hcz : dist c z ≤ e) :
closedBall c (e/2) ⊆ closedBall z (3*e) ∧
closedBall z (3*e) ⊆ closedBall c (10*e) := by
constructor
· intro w hw
have hdist := dist_triangle w c z
have hw' := mem_closedBall.mp hw
exact mem_closedBall.mpr (by linarith only [hdist,hw',hcz,he])
· intro w hw
have hdist := dist_triangle w z c
have hw' := mem_closedBall.mp hw
rw [dist_comm z c] at hdist
exact mem_closedBall.mpr (by linarith only [hdist,hw',hcz,he])
end CannonMagnification
end
end
end
end
end
end
end
end
end
end
end OAI
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