#!/usr/bin/env python3

from pantograph.server import Server

sketch = """
theorem add_comm_proved_formal_sketch : ∀ n m : Nat, n + m = m + n := by
   -- Consider some n and m in Nats.
   intros n m
   -- Perform induction on n.
   induction n with
   | zero =>
     -- Base case: When n = 0, we need to show 0 + m = m + 0.
     -- We have the fact 0 + m = m by the definition of addition.
     have h_base: 0 + m = m := sorry
     -- We also have the fact m + 0 = m by the definition of addition.
     have h_symm: m + 0 = m := sorry
     -- Combine facts to close goal
     sorry
   | succ n ih =>
     -- Inductive step: Assume n + m = m + n, we need to show succ n + m = m + succ n.
     -- By the inductive hypothesis, we have n + m = m + n.
     have h_inductive: n + m = m + n := sorry
     -- 1. Note we start with: Nat.succ n + m = m + Nat.succ n, so, pull the succ out from m + Nat.succ n on the right side from the addition using addition facts Nat.add_succ.
     have h_pull_succ_out_from_right: m + Nat.succ n = Nat.succ (m + n) := sorry
     -- 2. then to flip m + S n to something like S (n + m) we need to use the IH.
     have h_flip_n_plus_m: Nat.succ (n + m) = Nat.succ (m + n) := sorry
     -- 3. Now the n & m are on the correct sides Nat.succ n + m = Nat.succ (n + m), so let's use the def of addition to pull out the succ from the addition on the left using Nat.succ_add.
     have h_pull_succ_out_from_left: Nat.succ n + m = Nat.succ (n + m) := sorry
     -- Combine facts to close goal
     sorry
"""

if __name__ == '__main__':
    server = Server()
    state0, = server.load_sorry(sketch)
    print(state0)
