IndisputableMonolith.Verification.CoshPropertiesCert

   1import Mathlib
   2import IndisputableMonolith.Cost.FunctionalEquation
   3
   4namespace IndisputableMonolith
   5namespace Verification
   6namespace CoshProperties
   7
   8open IndisputableMonolith.Cost.FunctionalEquation
   9open Real
  10
  11/-!
  12# Cosh Properties Certificate: ODE and Initial Conditions
  13
  14This certificate packages the verification that cosh is indeed a solution
  15to the ODE H'' = H with initial conditions H(0) = 1, H'(0) = 0.
  16
  17## Key Results
  18
  191. **cosh(0) = 1**: Initial value condition
  202. **sinh(0) = 0**: cosh'(0) = sinh(0) = 0
  213. **cosh'' = cosh**: The ODE that cosh satisfies
  22
  23## Why this matters for the certificate chain
  24
  25The ODE uniqueness theorem says: if H'' = H with H(0) = 1, H'(0) = 0, then H = cosh.
  26
  27But we need to verify that cosh actually satisfies these conditions! Otherwise
  28the uniqueness theorem would be vacuously true (no solution exists).
  29
  30This certificate verifies:
  31- cosh is a solution (satisfies the ODE)
  32- cosh has the right initial conditions
  33- Therefore cosh is THE unique solution
  34
  35## Mathematical Content
  36
  37From the definitions:
  38- cosh t = (exp t + exp(-t)) / 2
  39- sinh t = (exp t - exp(-t)) / 2
  40
  41We verify:
  42- cosh(0) = (1 + 1) / 2 = 1
  43- sinh(0) = (1 - 1) / 2 = 0
  44- cosh' = sinh
  45- sinh' = cosh
  46- Therefore cosh'' = cosh
  47-/
  48
  49structure CoshPropertiesCert where
  50  deriving Repr
  51
  52/-- Verification predicate: cosh satisfies the ODE with correct initial conditions.
  53
  54This certifies:
  551. cosh(0) = 1 (initial value)
  562. cosh'(0) = sinh(0) = 0 (initial derivative)
  573. cosh'' = cosh (the ODE) -/
  58@[simp] def CoshPropertiesCert.verified (_c : CoshPropertiesCert) : Prop :=
  59  -- Initial conditions
  60  (Real.cosh 0 = 1 ∧ deriv (fun x => Real.cosh x) 0 = 0) ∧
  61  -- ODE: cosh'' = cosh
  62  (∀ t : ℝ, deriv (deriv (fun x => Real.cosh x)) t = Real.cosh t)
  63
  64/-- Top-level theorem: the certificate verifies. -/
  65@[simp] theorem CoshPropertiesCert.verified_any (c : CoshPropertiesCert) :
  66    CoshPropertiesCert.verified c := by
  67  constructor
  68  · exact cosh_initials
  69  · exact cosh_second_deriv_eq
  70
  71end CoshProperties
  72end Verification
  73end IndisputableMonolith
  74