The existence half of Picard-Lindelöf for finite-dimensional systems (Fin n → ℝ)
is now fully proven. The proof factors into:
1. exists_forward_solution — generic lemma building a forward solution on [0,∞)
for any Lipschitz vector field via Nat.rec over τ-intervals with
HasDerivWithinAt.union at boundaries and Nat.find indexing.
2. Time reversal via f_neg (x ↦ -f x) applied to exists_forward_solution gives
the backward solution.
3. γ_full splices both halves with case split at t=0.
4. The splice point t=0 is resolved via
hasDerivAt_iff_tendsto_slope_left_right, with the left slope obtained by
composing the negative-forward slope with y ↦ -y and an algebraic identity
γ_full(y) = γ_neg_fwd(-y) on y ≤ 0.
Previously 1 sorry (the existence existence) remained; it is now closed.
The uniqueness half was already proven via ODE_solution_unique_univ.
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