import Mathlib

-- Precondition definitions
@[reducible, simp]
def discrete_log_naive_precond (g h p : UInt64) : Prop :=
  -- !benchmark @start precond
  p.toNat > 1
  -- !benchmark @end precond

def is_discrete_log (g h p : Nat) (x : Nat) : Prop :=
  (Nat.pow g x) % p = h % p

-- !benchmark @start auxcode
-- !benchmark @end auxcode

-- Main function definitions
def discrete_log_naive (g h p : UInt64) (h_precond : discrete_log_naive_precond g h p) : Option UInt64 :=
  -- !benchmark @start code
  sorry
  -- !benchmark @end code

-- Postcondition definitions
@[reducible, simp]
def discrete_log_naive_postcond (g h p : UInt64) (result : Option UInt64) (h_precond : discrete_log_naive_precond g h p) : Prop :=
  -- !benchmark @start postcond
  match result with
  | some x => is_discrete_log g.toNat h.toNat p.toNat x.toNat ∧ x.toNat < p.toNat ∧
               ∀ k : Nat, k < x.toNat → ¬ is_discrete_log g.toNat h.toNat p.toNat k
  | none => ∀ k : Nat, k < p.toNat → ¬ is_discrete_log g.toNat h.toNat p.toNat k
  -- !benchmark @end postcond

-- !benchmark @start lemma
-- !benchmark @end lemma

-- Proof content
theorem discrete_log_naive_postcond_satisfied (g h p : UInt64) (h_precond : discrete_log_naive_precond g h p) :
    discrete_log_naive_postcond g h p (discrete_log_naive g h p h_precond) h_precond := by
  -- !benchmark @start proof
  sorry
  -- !benchmark @end proof