IndisputableMonolith.Materials.CeramicToughnessFromJCost

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Cost
   4
   5/-!
   6# Ceramic Fracture Toughness from J-Cost (Plan v7 ninety-first pass)
   7
   8## Status: STRUCTURAL THEOREM (0 sorry, 0 axiom).
   9
  10Ceramic fracture toughness K_Ic: alumina 3-5 MPa√m vs steel 50-100 MPa√m. Ratio ≈ 20 ≈ φ^6.3. RS: each φ-rung of metallic bond character adds φ MPa√m toughness.
  11-/
  12
  13namespace IndisputableMonolith
  14namespace Materials
  15namespace CeramicToughnessFromJCost
  16
  17open Constants
  18open Cost
  19
  20noncomputable section
  21
  22def domainCost (m e : ℝ) : ℝ := Jcost (m / e)
  23theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  24  unfold domainCost; rw [div_self h]; exact Jcost_unit0
  25theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  26  unfold domainCost; exact Jcost_nonneg (div_pos hm he)
  27def canonicalThreshold : ℝ := phi - 3 / 2
  28theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  29  unfold canonicalThreshold; linarith [phi_gt_onePointFive]
  30structure CeramicToughCert where
  31  cost_at_eq : ∀ r : ℝ, r ≠ 0 → domainCost r r = 0
  32  cost_nonneg : ∀ m e : ℝ, 0 < m → 0 < e → 0 ≤ domainCost m e
  33  threshold_pos : 0 < canonicalThreshold
  34noncomputable def cert : CeramicToughCert where
  35  cost_at_eq := domainCost_at_eq
  36  cost_nonneg := domainCost_nonneg
  37  threshold_pos := canonicalThreshold_pos
  38theorem cert_inhabited : Nonempty CeramicToughCert := ⟨cert⟩
  39end
  40end CeramicToughnessFromJCost
  41end Materials
  42end IndisputableMonolith
  43