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9b6e014 | 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 43 44 45 46 47 | import OAI.Probability.ContinuumTransition.TripleState
namespace OAI
section
open scoped Topology
open Set Filter Metric
noncomputable section
namespace ContinuumTemperature
structure ConstructionCap (t M : ℝ) : Prop where
t_large : 1 < t
mass_pos : 0 < M
strong : 140608*kernelHeight*M ≤ t
gap : 1024*M < t
lemma ConstructionCap.weak {t M m : ℝ} (h : ConstructionCap t M) (hm : m ≤ M) :
17576*kernelHeight*m ≤ t := by
calc
17576*kernelHeight*m ≤ 17576*kernelHeight*M := mul_le_mul_of_nonneg_left hm (by positivity [kernelHeight_pos])
_ ≤ 140608*kernelHeight*M := by nlinarith only [mul_pos kernelHeight_pos h.mass_pos]
_ ≤ t := h.strong
lemma ConstructionCap.small {t M m : ℝ} (h : ConstructionCap t M) (hm : m ≤ M) :
2197*t*kernelHeight*m ≤ (t^2/8)/8 := by
apply strong_smallness (lt_trans zero_lt_one h.t_large)
exact (mul_le_mul_of_nonneg_left hm (by positivity [kernelHeight_pos])).trans h.strong
lemma ConstructionCap.lt_gap {t M a : ℝ} (h : ConstructionCap t M) (ha : a ≤ M) :
1024*a < t := (mul_le_mul_of_nonneg_left ha (by norm_num)).trans_lt h.gap
lemma exists_constructionCap {M : ℝ} (hM : 0 < M) : ∃ T : ℝ, ConstructionCap T M := by
obtain ⟨T,hT⟩ := exists_gt (max 1 (max (140608*kernelHeight*M) (1024*M)))
refine ⟨T,?_,hM,?_,?_⟩
· exact (le_max_left _ _).trans_lt hT
· exact ((le_max_left _ _).trans (le_max_right _ _)).trans hT.le
· exact ((le_max_right _ _).trans (le_max_right _ _)).trans_lt hT
lemma ConstructionCap.mono_t {t T M : ℝ} (h : ConstructionCap T M) (ht : T ≤ t) :
ConstructionCap t M := ⟨h.t_large.trans_le ht,h.mass_pos,h.strong.trans ht,h.gap.trans_le ht⟩
end ContinuumTemperature
end
end
end OAI
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