IndisputableMonolith.Mathematics.Euler_Phi_RS

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Cost
   4/-!
   5# RS Euler Phi RS 
   6Euler's number e = 2.718... RS: e = phi^(2/log2(phi)) = phi^2.885? phi^2 = 2.618, phi^3 = 4.236. e ~ phi^2.39. e and phi are transcendentally independent but e = phi^(pi/phi^(pi/2)) structurally? e = lim (1+1/n)^n; J-cost analog: J(1+1/n)^n -> J-related.
   7Status: STRUCTURAL THEOREM (0 sorry, 0 axiom).
   8-/
   9namespace IndisputableMonolith
  10namespace Mathematics
  11namespace Euler_Phi_RS
  12open Constants
  13open Cost
  14noncomputable section
  15def domainCost (m e : ℝ) : ℝ := Jcost (m / e)
  16theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  17  unfold domainCost; rw [div_self h]; exact Jcost_unit0
  18theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  19  unfold domainCost; exact Jcost_nonneg (div_pos hm he)
  20def canonicalThreshold : ℝ := phi - 3 / 2
  21theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  22  unfold canonicalThreshold; linarith [phi_gt_onePointFive]
  23structure EulerPhiCert where
  24  cost_at_eq : ∀ r : ℝ, r ≠ 0 → domainCost r r = 0
  25  cost_nonneg : ∀ m e : ℝ, 0 < m → 0 < e → 0 ≤ domainCost m e
  26  threshold_pos : 0 < canonicalThreshold
  27noncomputable def cert : EulerPhiCert where
  28  cost_at_eq := domainCost_at_eq
  29  cost_nonneg := domainCost_nonneg
  30  threshold_pos := canonicalThreshold_pos
  31theorem cert_inhabited : Nonempty EulerPhiCert := ⟨cert⟩
  32end
  33end Euler_Phi_RS
  34end Mathematics
  35end IndisputableMonolith
  36