rcases hf₄ with ⟨pf₄, hpf₄⟩
case mk.intro.mk.intro.mk.intro
L : Language
inst✝⁹ : L.ORing
ξ : Type v
n : ℕ
M : Type u_1
inst✝⁸ : Zero M
inst✝⁷ : One M
inst✝⁶ : Add M
inst✝⁵ : Mul M
inst✝⁴ : LT M
inst✝³ : M ⊧ₘ* 𝐏𝐀⁻
inst✝² : Structure L M
inst✝¹ : Structure.ORing L M
inst✝ : Structure.Monotone L M
Γ✝ : HierarchySymbol
k✝ : ℕ
P✝ Q : (Fin k✝ → M) → Prop
m k : ℕ
P : M → M → M → M → Prop
f₁ f₂ f₃ f₄ : (Fin k → M) → M
hf₄ : DefinableFunction L (𝚺, m + 1) f₄
Γ : SigmaPiDelta
hP : DefinableRel₄ L (Γ, m + 1) P
pf₁ : HSemiformula L M (k + 1) (𝚺, m + 1)
hpf₁ : DefinedWithParam (fun v => v 0 = f₁ fun x => v x.succ) pf₁
pf₂ : HSemiformula L M (k + 1) (𝚺, m + 1)
hpf₂ : DefinedWithParam (fun v => v 0 = f₂ fun x => v x.succ) pf₂
pf₃ : HSemiformula L M (k + 1) (𝚺, m + 1)
hpf₃ : DefinedWithParam (fun v => v 0 = f₃ fun x => v x.succ) pf₃
⊢ Definable L (Γ, m + 1) fun v => P (f₁ v) (f₂ v) (f₃ v) (f₄ v)
case mk.intro.mk.intro.mk.intro.mk.intro
L : Language
inst✝⁹ : L.ORing
ξ : Type v
n : ℕ
M : Type u_1
inst✝⁸ : Zero M
inst✝⁷ : One M
inst✝⁶ : Add M
inst✝⁵ : Mul M
inst✝⁴ : LT M
inst✝³ : M ⊧ₘ* 𝐏𝐀⁻
inst✝² : Structure L M
inst✝¹ : Structure.ORing L M
inst✝ : Structure.Monotone L M
Γ✝ : HierarchySymbol
k✝ : ℕ
P✝ Q : (Fin k✝ → M) → Prop
m k : ℕ
P : M → M → M → M → Prop
f₁ f₂ f₃ f₄ : (Fin k → M) → M
Γ : SigmaPiDelta
hP : DefinableRel₄ L (Γ, m + 1) P
pf₁ : HSemiformula L M (k + 1) (𝚺, m + 1)
hpf₁ : DefinedWithParam (fun v => v 0 = f₁ fun x => v x.succ) pf₁
pf₂ : HSemiformula L M (k + 1) (𝚺, m + 1)
hpf₂ : DefinedWithParam (fun v => v 0 = f₂ fun x => v x.succ) pf₂
pf₃ : HSemiformula L M (k + 1) (𝚺, m + 1)
hpf₃ : DefinedWithParam (fun v => v 0 = f₃ fun x => v x.succ) pf₃
pf₄ : HSemiformula L M (k + 1) (𝚺, m + 1)
hpf₄ : DefinedWithParam (fun v => v 0 = f₄ fun x => v x.succ) pf₄
⊢ Definable L (Γ, m + 1) fun v => P (f₁ v) (f₂ v) (f₃ v) (f₄ v)