import Axiom

namespace dn8bd2yh

 def boundedAdaptationUpdate (state target eta : Int) : Int :=
  if eta = 0 then state else target

 theorem bounded_update_stays_between_state_and_target
    (state target eta : Int)
    (h_order : state ≤ target) :
    state ≤ boundedAdaptationUpdate state target eta ∧
      boundedAdaptationUpdate state target eta ≤ target := by
  by_cases h : eta = 0
  · simp [boundedAdaptationUpdate, h, h_order]
  · simp [boundedAdaptationUpdate, h, h_order]

 theorem zero_adaptation_rate_preserves_state
    (state target : Int) :
    boundedAdaptationUpdate state target 0 = state := by
  simp [boundedAdaptationUpdate]

 def continuationSufficient
    {History State Outcome : Type}
    (representation : History → State)
    (outcome : History → Outcome)
    (decoder : State → Outcome) : Prop :=
  ∀ history, outcome history = decoder (representation history)

 theorem sufficient_representation_preserves_continuation
    {History State Outcome : Type}
    (representation : History → State)
    (outcome : History → Outcome)
    (decoder : State → Outcome)
    (h_sufficient : continuationSufficient representation outcome decoder)
    (h₁ h₂ : History)
    (h_same : representation h₁ = representation h₂) :
    outcome h₁ = outcome h₂ := by
  rw [h_sufficient h₁, h_sufficient h₂, h_same]

def dynamical_theorem_boundaryPayload := "The bounded-update theorem uses a discrete Int model in which eta zero retains the current state and any nonzero eta up to one adopts the target. It proves only a bounded one-step result. It does not prove that large learning rates cause thrashing, that targets are stationary, or that the model describes an athlete or an AI system without measurement. The continuation theorem proves preservation only when the declared factorization premise holds; it does not prove that a chosen representation is sufficient."
axiom dynamical_theorem_boundary_auth : Axiom.AuthenticatedTerm ("hitchhiker2thefuture", dynamical_theorem_boundaryPayload, 1789062787)
def dynamical_theorem_boundary : Axiom.Statement "hitchhiker2thefuture" dynamical_theorem_boundaryPayload 1789062787 := ⟨dynamical_theorem_boundary_auth⟩

end dn8bd2yh