{
  "foldl_add_reverse": "```lean\ntheorem foldl_add_reverse (l : List Score) (init : Score) :\n    l.reverse.foldl Score.add init = l.foldl Score.add init := by\n  induction' l with x xs ih generalizing init\n  \u00b7 rfl\n  \u00b7 simp [List.reverse_cons, List.foldl_append, List.foldl_cons, List.foldl_nil]\n    have h : \u2200 (xs : List Score) (s : Score), (xs.reverse.foldl Score.add s) = (xs.foldl Score.add s) := by\n      intro xs s\n      exact ih s\n    rw [h]\n    simp\n    have helper : \u2200 (xs : List Score) (s t : Score), (xs.foldl Score.add (s.add t)) = (xs.foldl Score.add s).add t := by\n      intro xs\n      induction' xs with y ys ih' generalizing s t\n      \u00b7 simp\n      \u00b7 simp [ih' (s.add t) y, Score.add_assoc]\n    rw [helper xs init x]\n    simp [Score.add_comm, Score.add_assoc]",
  "pathCost_reverse": "```lean\ntheorem pathCost_reverse (p : Path) : pathCost (p.reversePath) = pathCost p := by\n  unfold pathCost reversePath\n  simp\n  induction p generalizing ?_ with\n  | nil => rfl\n  | cons e es ih =>\n    simp [pathCost, List.foldl, List.reverseAux, List.map, AdmissibilityEdge.reverse]\n    rw [ih]\n    simp [pathCost, List.foldl]\n    rfl\n```",
  "shellMass_max_at_midpoint": "```lean\ntheorem shellMass_max_at_midpoint (k : Nat) :\n    let n := k * k + k\n    shellMass n = k * (k + 1) := by\n  intro n\n  unfold shellMass\n  have h_sqrt : Nat.sqrt n = k := by\n    apply Nat.sqrt_eq_iff_mul_self_eq\n    \u00b7 have : k * k \u2264 n := by\n        nlinarith\n      exact this\n    \u00b7 have : n < (k + 1) * (k + 1) := by\n        nlinarith\n      exact this\n  rw [h_sqrt]\n  have h_a : n - k * k = k := by\n    unfold n\n    nlinarith\n  have h_b : (k + 1) * (k + 1) - n = k + 1 := by\n    unfold n\n    nlinarith\n  rw [h_a, h_b]\n  ring\n```"
}