Theorem: Super.preimage_eq (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
o : OnePreimage s
n : ℕ
⊢ (f c)^[n] z = a ↔ z = a
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
o : OnePreimage s
⊢ (f c)^[0] z = a ↔ z = a

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
o : OnePreimage s
n : ℕ
h : (f c)^[n] z = a ↔ z = a
⊢ (f c)^[n + 1] z = a ↔ z = a

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
o : OnePreimage s
⊢ (f c)^[0] z = a ↔ z = a

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
o : OnePreimage s
n : ℕ
h : (f c)^[n] z = a ↔ z = a
⊢ (f c)^[n + 1] z = a ↔ z = a
Tactic:
simp only [Function.iterate_zero_apply]
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
o : OnePreimage s
n : ℕ
h : (f c)^[n] z = a ↔ z = a
⊢ (f c)^[n + 1] z = a ↔ z = a

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
o : OnePreimage s
n : ℕ
h : (f c)^[n] z = a ↔ z = a
⊢ (f c)^[n + 1] z = a ↔ z = a
Tactic:
simp only [Function.iterate_succ_apply', s.preimage_eq', h]
State after:
no goals