Problem: llrbt_flipcolor

Your task is to implement and verify the flip_colors algorithm for a Left-Leaning Red-Black Tree (LLRBT) in Lean. Verification Specification: (1) Preconditions: The inputs (node, left child, right child) must form a valid local cluster where the node has a different color from its children, and both children have the same color. (2) Postconditions: Functional Correctness (view/set is preserved) and Invariant Maintenance (BST order is preserved using is_llrbt predicate, Black Height is preserved meaning the flip effectively pushes redness up or down without changing the black count on any path, and the colors of the node and its children are inverted: if node was Black and children Red, node becomes Red and children Black, and vice-versa).


import Mathlib

inductive Color
| Red
| Black
deriving Inhabited, BEq

inductive Node
| Empty : Node
| Tree  (color : Color) (val : Int) (left : Node) (right : Node) : Node
deriving Inhabited

def view (t : Node) : Set Int :=
  match t with
  | Node.Empty => ∅
  | Node.Tree _ v l r => view l ∪ view r ∪ {v}

def is_red (t : Node) : Bool :=
  match t with
  | Node.Tree Color.Red _ _ _ => true
  | _ => false

def get_color (t : Node) : Color :=
  match t with
  | Node.Tree c _ _ _ => c
  | Node.Empty => Color.Black -- Empty nodes are black

def is_bst (t : Node) : Prop :=
  match t with
  | Node.Empty => True
  | Node.Tree _ v l r =>
    (∀ x ∈ view l, x < v) ∧
    is_bst l ∧
    (∀ x ∈ view r, x > v) ∧
    is_bst r

-- Black Height: Returns -1 if invalid
def black_height (t : Node) : Int :=
  match t with
  | Node.Empty => 1
  | Node.Tree c _ l r =>
    let lh := black_height l
    let rh := black_height r
    if lh ≠ -1 ∧ rh ≠ -1 ∧ lh = rh then
      if c == Color.Black then lh + 1 else lh
    else
      -1

def satisfies_red_props (t : Node) : Prop :=
  match t with
  | Node.Empty => True
  | Node.Tree _ _ l r =>
    satisfies_red_props l ∧
    satisfies_red_props r ∧
    -- Rule: No Right-Leaning Red
    (¬ is_red r) ∧
    -- Rule: No Double Red on Left
    (match l with
     | Node.Tree Color.Red _ ll _ => ¬ is_red ll
     | _ => True)

-- Main Invariant
def is_llrbt (t : Node) : Prop :=
  is_bst t ∧
  black_height t ≠ -1 ∧
  satisfies_red_props t

def get_left (t : Node) : Option Node :=
  match t with | Node.Tree _ _ l _ => some l | _ => none

def get_right (t : Node) : Option Node :=
  match t with | Node.Tree _ _ _ r => some r | _ => none

def get_val (t : Node) : Option Int :=
  match t with | Node.Tree _ v _ _ => some v | _ => none

def is_not_empty (t : Node) : Prop :=
  match t with | Node.Empty => False | _ => True

-- Precondition definitions
@[reducible, simp]
def flip_colors_precond (t : Node) : Prop :=
  -- !benchmark @start precond
  -- t is not empty
  is_not_empty t ∧
  is_bst t ∧
  match t with
  | Node.Tree c _ l r =>
    is_not_empty l ∧ is_not_empty r ∧
    (get_color l = get_color r) ∧
    (c ≠ get_color l)
  | _ => False
  -- !benchmark @end precond

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

-- Main function definition
def flip_colors (t : Node) 
    (h_precond : flip_colors_precond t) : Node :=
  -- !benchmark @start code
  sorry
  -- !benchmark @end code

-- Postcondition definitions
@[reducible, simp]
def flip_colors_postcond (t : Node) (result : Node)
    (_ : flip_colors_precond t) : Prop :=
  -- !benchmark @start postcond
  view result = view t ∧
  is_bst result ∧ -- Requirement 1: BST structure preserved
  black_height result = black_height t ∧ -- Requirement 2: Balance preserved (Critical)
  match result, t with
  | Node.Tree c' _ l' r', Node.Tree c _ l r =>
      c' ≠ c ∧
      get_color l' ≠ get_color l ∧
      get_color r' ≠ get_color r
  | _, _ => False
  -- !benchmark @end postcond

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

-- Proof content
theorem flip_colors_postcond_satisfied (t : Node)
    (h_precond : flip_colors_precond t) :
    flip_colors_postcond t (flip_colors t h_precond) h_precond := by
  -- !benchmark @start proof
  sorry
  -- !benchmark @end proof