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f205175 | 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 | import Mathlib
import OAI.Probability.CriticalPercolation.Gluing.RelayScales
namespace OAI
section
section
noncomputable section
open Set Function MeasureTheory ProbabilityTheory
namespace CriticalPercolation.BondGraph
variable {V E : Type*} (G : BondGraph V E)
lemma source_outside_relay_collar (t z : V → ℝ) (σ e ρ P : ℝ) (L R m i : ℕ)
(he : 0<e) (hρ : 0 ≤ ρ) (hi : i ≤ L) (hPi : 1 ≤ P+i)
(heL : (L+3)*e<1/100) (hr : (R+m+1)*ρ<e/10)
(ht : ∀ ⦃u v⦄, G.fullGraph.Adj u v → |t u-t v| ≤ ρ)
(hz : ∀ ⦃u v⦄, G.fullGraph.Adj u v → |z u-z v| ≤ ρ)
{a : V} (ha : t a ≤ m*ρ-P) :
a∉G.vertexCollar (corridorRelay t z σ e ρ L i) R := by
intro hin
obtain ⟨hρe,_,_,_,hlast⟩ := relay_coordinate_scale L R m e ρ he hρ heL hr
have hs := G.vertexCollar_mono (corridorRelay_subset t z σ e ρ hρe L i hi) R hin
have hb := (G.relay_collar_bounds t z σ e ρ hρ ht hz R i hs).1
have hi' : (i:ℝ) ≤ L := by exact_mod_cast hi
have hile : (i+1)*e ≤ (L+1)*e := mul_le_mul_of_nonneg_right (by linarith) he.le
have hb' := (abs_le.mp hb).1
nlinarith
end CriticalPercolation.BondGraph
end
end
end
end OAI
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