Problem: ternarysearchtree_search

Your task is to implement the Search operation for a Ternary Search Tree (TST) and verify its correctness in Dafny. A TST is a hybrid data structure that combines the time efficiency of Tries with the space efficiency of Binary Search Trees. Unlike a standard Trie, which wastes memory by using a large array for every possible child (e.g., 256 slots per node), a TST node stores just one character (val) and three pointers: left, right, and mid. The left and right children form a BST for "sibling" characters at the current position, while the mid child acts as a Trie link to the next character in the sequence. The search logic must respect this three-way branching: (1) Horizontal Traversal (BST Logic): If the current character in the key is less than the node's val, recurse to left. If it is greater than, recurse to right. In both cases, do not consume the character from the key. (2) Vertical Traversal (Trie Logic): If the current character is equal to the node's val, the character is matched. Consume it and recurse to mid with the remaining key. Specifically, you must prove: (1) Functional Correctness: The function returns true if and only if the key exists in the tree's abstract view. (2) Traversal Invariant: The implementation correctly navigates the BST structure for mismatches and the Trie structure for matches.

// Following is the block for necessary definitions
// <preamble>
datatype Option<T> = Some(value: T) | None

datatype Node = Node(
    val: int,            // The character at this node (u8)
    is_end: bool,        // Does a key end here?
    left: Option<Node>,  // Char < val
    mid: Option<Node>,   // Char == val (next char)
    right: Option<Node>  // Char > val
) {
    // A node is "empty" if it marks no key and has no children.
    // Used for pruning (cleanup).
    ghost predicate is_empty() {
        && !this.is_end
        && this.left.None?
        && this.mid.None?
        && this.right.None?
    }

    // Structural Integrity + BST Invariants + Pruning
    // This is the core correctness predicate.
    ghost predicate well_formed() 
        decreases this
    {
        // 0. Type constraint for u8
        && 0 <= this.val < 256
        
        // 1. Recursive structural validity
        // 2. Pruning: No child should be "empty"
        // 3. BST Property: 
        //    - All keys in 'left' must start with char < this.val
        //    - All keys in 'right' must start with char > this.val
        
        // Check Left Child
        && (match this.left {
            case Some(l) => 
                && l.well_formed()
                && !l.is_empty()
                // The BST Invariant for Left:
                && (forall s | s in l.view() :: |s| > 0 && s[0] < this.val)
            case None => true
        })
        // Check Right Child
        && (match this.right {
            case Some(r) => 
                && r.well_formed()
                && !r.is_empty()
                // The BST Invariant for Right:
                && (forall s | s in r.view() :: |s| > 0 && s[0] > this.val)
            case None => true
        })
        // Check Middle Child (Trie Property)
        && (match this.mid {
            case Some(m) => 
                && m.well_formed()
                && !m.is_empty()
                // No character constraints on mid relative to this.val, 
                // because mid represents the *next* character.
            case None => true
        })
    }

    // Recursive definition of containment.
    // Note: TST nodes typically store non-empty paths relative to the parent's branching.
    ghost function contains(key: seq<int>): bool
        decreases this
    {
        if |key| == 0 then
            // A node always implies a character; empty string cannot reside "in" a node in this encoding
            false 
        else
            var c := key[0];
            if c < this.val then
                // Delegate to left, key remains same
                match this.left {
                    case Some(l) => l.contains(key)
                    case None => false
                }
            else if c > this.val then
                // Delegate to right, key remains same
                match this.right {
                    case Some(r) => r.contains(key)
                    case None => false
                }
            else
                // c == this.val. Match found.
                if |key| == 1 then
                    this.is_end
                else
                    // Continue to mid with the rest of the key
                    match this.mid {
                        case Some(m) => m.contains(key[1..])
                        case None => false
                    }
    }

    // View as a Set of byte sequences
    // Using iset (infinite set) for abstraction, though the actual set is finite.
    ghost function view(): iset<seq<int>> {
        iset key: seq<int> | this.contains(key)
    }
}

ghost function opt_view(node: Option<Node>): iset<seq<int>> {
    match node {
        case Some(n) => n.view()
        case None => iset{}
    }
}

ghost predicate opt_well_formed(node: Option<Node>) {
    match node {
        case Some(n) => n.well_formed() && !n.is_empty()
        case None => true
    }
}
// </preamble>

// Following is the block for potential helper specifications
// <helpers>

// </helpers>

// Following is the block for proofs of lemmas
// <proofs>

// </proofs>

// Following is the block for the main specification
// <spec>
method search(tree: Option<Node>, key: seq<int>) returns (res: bool)
    requires opt_well_formed(tree)
    // Constraint: Input key elements must represent bytes (u8)
    requires forall k :: k in key ==> 0 <= k < 256
    ensures res == (key in opt_view(tree))
// </spec>
// <code>
{
    // Implement and verify the search function for Ternary Search Tree (TST).
}
// </code>

method main() {}