IndisputableMonolith.Verification.JcostStrictCert

   1import Mathlib
   2import IndisputableMonolith.Cost
   3
   4namespace IndisputableMonolith
   5namespace Verification
   6namespace JcostStrict
   7
   8open IndisputableMonolith.Cost
   9
  10/-!
  11# Jcost Strict Positivity Certificate
  12
  13This certificate packages the strict positivity result for Jcost that
  14strengthens the minimum uniqueness story.
  15
  16## Key Result
  17
  18**Strict Positivity**: J(x) > 0 for x ≠ 1 and x > 0
  19
  20## Why this matters for the certificate chain
  21
  22We already know:
  23- J(x) ≥ 0 for x > 0 (certified in JcostAxiomsCert)
  24- J(1) = 0 (certified in JcostAxiomsCert)
  25
  26This certificate adds the strict version:
  27- J(x) = 0 **only** at x = 1 (strict positivity elsewhere)
  28
  29This proves that x = 1 is the **unique global minimum**.
  30
  31## Mathematical Content
  32
  33From J(x) = (x-1)²/(2x):
  34- Numerator (x-1)² > 0 when x ≠ 1 (strict positivity of squares)
  35- Denominator 2x > 0 when x > 0
  36- Therefore J(x) > 0 when x ≠ 1 and x > 0
  37-/
  38
  39structure JcostStrictCert where
  40  deriving Repr
  41
  42/-- Verification predicate: Jcost has strict positivity away from 1.
  43
  44This certifies: J(x) > 0 for x ≠ 1 and x > 0 -/
  45@[simp] def JcostStrictCert.verified (_c : JcostStrictCert) : Prop :=
  46  -- Strict positivity: J(x) > 0 for x ≠ 1, x > 0
  47  ∀ x : ℝ, 0 < x → x ≠ 1 → 0 < Jcost x
  48
  49/-- Top-level theorem: the certificate verifies. -/
  50@[simp] theorem JcostStrictCert.verified_any (c : JcostStrictCert) :
  51    JcostStrictCert.verified c := by
  52  intro x hx hx1
  53  exact Jcost_pos_of_ne_one x hx hx1
  54
  55end JcostStrict
  56end Verification
  57end IndisputableMonolith
  58