Theorem: Super.bottcherNear_eqn_iter (Ray/Dynamics/BottcherNearM.lean)

State before:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
m : (c, z) ∈ s.near
n : ℕ
⊢ s.bottcherNear c ((f c)^[n] z) = s.bottcherNear c z ^ d ^ n
Tactic:
induction' n with n h
State after:
case zero
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
m : (c, z) ∈ s.near
⊢ s.bottcherNear c ((f c)^[0] z) = s.bottcherNear c z ^ d ^ 0

case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
m : (c, z) ∈ s.near
n : ℕ
h : s.bottcherNear c ((f c)^[n] z) = s.bottcherNear c z ^ d ^ n
⊢ s.bottcherNear c ((f c)^[n + 1] z) = s.bottcherNear c z ^ d ^ (n + 1)

State before:
case zero
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
m : (c, z) ∈ s.near
⊢ s.bottcherNear c ((f c)^[0] z) = s.bottcherNear c z ^ d ^ 0

case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
m : (c, z) ∈ s.near
n : ℕ
h : s.bottcherNear c ((f c)^[n] z) = s.bottcherNear c z ^ d ^ n
⊢ s.bottcherNear c ((f c)^[n + 1] z) = s.bottcherNear c z ^ d ^ (n + 1)
Tactic:
simp only [Function.iterate_zero_apply, pow_zero, pow_one]
State after:
case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
m : (c, z) ∈ s.near
n : ℕ
h : s.bottcherNear c ((f c)^[n] z) = s.bottcherNear c z ^ d ^ n
⊢ s.bottcherNear c ((f c)^[n + 1] z) = s.bottcherNear c z ^ d ^ (n + 1)

State before:
case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
m : (c, z) ∈ s.near
n : ℕ
h : s.bottcherNear c ((f c)^[n] z) = s.bottcherNear c z ^ d ^ n
⊢ s.bottcherNear c ((f c)^[n + 1] z) = s.bottcherNear c z ^ d ^ (n + 1)
Tactic:
simp only [Function.iterate_succ_apply', s.bottcherNear_eqn (s.iter_stays_near m n), h, ←
  pow_mul, ← pow_succ]
State after:
no goals