Orbit succ'

-- Thm stub generated from Novelty/EscapeCriterionIteration.lean
import Mathlib
import Definitions.Def_Novelty_EscapeCriterionIteration
import Definitions.Def_Novelty_MandelbrotQuadraticEscape

/-!
# A formally specified escape-time test for the quadratic family

This file strengthens the escape estimates of `Novelty.MandelbrotQuadraticEscape`
(where divergence of the *critical* orbit is proved only under the a priori hypothesis
`2 < ‖c‖`) into a **complete escape criterion for arbitrary orbits** of `f_c(z) = z² + c`:

* `escapeRadius c = max 2 ‖c‖` is the standard escape radius.
* `escape_norm_growth`: once an orbit point strictly exceeds the escape radius, the whole
  forward orbit stays in the escaping region and grows geometrically,
  `(‖z‖ - 1)^n * ‖z‖ ≤ ‖orbit c z n‖`, with ratio `‖z‖ - 1 > 1`.
* `escape_tendsto_atTop`: hence the orbit norm tends to infinity — crossing the escape
  radius *once* certifies divergence.
* `escape_time_bound`: an **effective** escape time: an explicit number of iterations after
  which the orbit provably exceeds a prescribed threshold `B`.
* `bounded_iff_never_escapes`: soundness *and* completeness of the escape-time test:
  an orbit is bounded iff it never crosses the escape radius.
* `mem_Mandelbrot_iff`: `c ∈ M ↔ ∀ n, ‖critOrbit c n‖ ≤ 2` — the radius-`2` test used by
  every Mandelbrot renderer, now a theorem rather than a heuristic.
* `escapeRadius_sharp`: the radius `2` cannot be lowered (witness `c = -2`).
* Topological payoff: `Mandelbrot_eq_iInter`, `isClosed_Mandelbrot`, `isCompact_Mandelbrot`:
  the dynamical estimate converts the escape-time algorithm into the statement that `M`
  is a nested intersection of closed test sets, hence compact.
-/

open EscapeCriterion

open Filter MandelbrotEscape
open scoped Topology

/-! ## Orbits of arbitrary starting points -/

theorem EscapeCriterion.orbit_succ_p(c z : ℂ) (n : ℕ) : orbit c z (n + 1) = qmap c (orbit c z n) := by sorry