constructor
σ : Instantiation
σ' : VarName → VarName
a1_left : ↑σ ∘ σ' = id
a1_right : σ' ∘ ↑σ = id
σ_inv : Instantiation := ⟨σ', ⋯⟩
⊢ ↑σ ∘ ↑σ_inv = id ∧ ↑σ_inv ∘ ↑σ = id
case left
σ : Instantiation
σ' : VarName → VarName
a1_left : ↑σ ∘ σ' = id
a1_right : σ' ∘ ↑σ = id
σ_inv : Instantiation := ⟨σ', ⋯⟩
⊢ ↑σ ∘ ↑σ_inv = id

case right
σ : Instantiation
σ' : VarName → VarName
a1_left : ↑σ ∘ σ' = id
a1_right : σ' ∘ ↑σ = id
σ_inv : Instantiation := ⟨σ', ⋯⟩
⊢ ↑σ_inv ∘ ↑σ = id