Problem: rod_cutting

Your task is to implement the Rod Cutting algorithm and verify its correctness in Lean. Given a rod of length n and a table of prices (where prices[i] is the price of a piece of length i+1), determine the maximum revenue obtainable by cutting the rod given the prices. Postconditions: Upper Bound (result ≥ revenue of any valid cut strategy) and Existence (result = revenue of some valid cut strategy). Valid strategy means cut lengths sum to n.


import Mathlib

def sum_lengths (cuts : List Nat) : Nat :=
  cuts.foldl (· + ·) 0

def get_price (prices : List Nat) (len : Nat) : Nat :=
  if len > 0 then prices.getD (len - 1) 0 else 0

def calculate_revenue (cuts : List Nat) (prices : List Nat) : Nat :=
  cuts.foldl (fun acc len => acc + get_price prices len) 0

def is_valid_strategy (cuts : List Nat) (n : Nat) (prices : List Nat) : Prop :=
  (∀ len ∈ cuts, len > 0 ∧ len - 1 < prices.length) ∧
  sum_lengths cuts = n

-- Precondition definitions
@[reducible, simp]
def rod_cutting_precond (n : Nat) (prices : List Nat) : Prop :=
  -- !benchmark @start precond
  prices.length ≥ n ∧
  n ≤ 1000 ∧
  (∀ i, i < prices.length → prices.getD i 0 ≤ 10000)
  -- !benchmark @end precond

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

-- Main function definition
def rod_cutting (n : Nat) (prices : List Nat)
    (h_precond : rod_cutting_precond n prices) : Nat :=
  -- !benchmark @start code
  sorry
  -- !benchmark @end code

-- Postcondition definitions
@[reducible, simp]
def rod_cutting_postcond (n : Nat) (prices : List Nat) (result : Nat)
    (_ : rod_cutting_precond n prices) : Prop :=
  -- !benchmark @start postcond
  (∀ cuts, is_valid_strategy cuts n prices → calculate_revenue cuts prices ≤ result) ∧
  (∃ cuts, is_valid_strategy cuts n prices ∧ calculate_revenue cuts prices = result)
  -- !benchmark @end postcond

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

-- Proof content
theorem rod_cutting_postcond_satisfied (n : Nat) (prices : List Nat)
    (h_precond : rod_cutting_precond n prices) :
    rod_cutting_postcond n prices (rod_cutting n prices h_precond) h_precond := by
  -- !benchmark @start proof
  sorry
  -- !benchmark @end proof