cases e₂ <;> simp [rcns] at hp ⊢
α : Type
inst✝ : Proper α
s s₁ s₂ s₁✝ s₂✝ : State α
e : s₁✝ ↓ᵢ s₂✝
e₂ : s₁✝ ↓ᵢ+ s₂
hp : rcns (single e) ~ rcns e₂
⊢ s₂✝.rtr = s₂.rtr ∧ s₂✝.events = s₂.events
case single
α : Type
inst✝ : Proper α
s s₁ s₂ s₁✝ s₂✝ : State α
e : s₁✝ ↓ᵢ s₂✝
a✝ : s₁✝ ↓ᵢ s₂
hp : rcn e = rcn a✝
⊢ s₂✝.rtr = s₂.rtr ∧ s₂✝.events = s₂.events

case trans
α : Type
inst✝ : Proper α
s s₁ s₂ s₁✝ s₂✝¹ : State α
e : s₁✝ ↓ᵢ s₂✝¹
s₂✝ : State α
a✝¹ : s₁✝ ↓ᵢ s₂✝
a✝ : s₂✝ ↓ᵢ+ s₂
hp : rcn e = rcn a✝¹ ∧ [] = rcns a✝
⊢ s₂✝¹.rtr = s₂.rtr ∧ s₂✝¹.events = s₂.events