Theorem: Super.potential_eqn_iter (Ray/Dynamics/Potential.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
n : ℕ
⊢ s.potential c ((f c)^[n] z) = s.potential 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
⊢ s.potential c ((f c)^[0] z) = s.potential 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
n : ℕ
h : s.potential c ((f c)^[n] z) = s.potential c z ^ d ^ n
⊢ s.potential c ((f c)^[n + 1] z) = s.potential 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
⊢ s.potential c ((f c)^[0] z) = s.potential c z ^ d ^ 0
Tactic:
simp only [Function.iterate_zero, id, pow_zero, pow_one]
State after:
no goals

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
n : ℕ
h : s.potential c ((f c)^[n] z) = s.potential c z ^ d ^ n
⊢ s.potential c ((f c)^[n + 1] z) = s.potential c z ^ d ^ (n + 1)
Tactic:
simp only [Function.iterate_succ', Super.potential_eqn, h, ← pow_mul, ← pow_succ,
  Function.comp]
State after:
no goals