match Γ with
| 𝚺 =>
  rcases hP with ⟨p, hp⟩; rcases hf with ⟨pf, hpf⟩
  exact ⟨(pf ⋏ (p.rew [→ #0])).ex, by intro v; simp [hp.df.iff, hpf.df.iff]⟩
| 𝚷 =>
  rcases hP with ⟨p, hp⟩; rcases hf with ⟨pf, hpf⟩
  exact ⟨(pf.negSigma ⋎ (p.rew [→ #0])).all, by intro v; simp [hp.df.iff, hpf.df.iff, ←imp_iff_not_or]⟩
| 𝚫 =>
  rcases hP with ⟨p, hp⟩; rcases hf with ⟨pf, hpf⟩
  exact of_sigma_of_pi
    ⟨(pf ⋏ (p.sigma.rew [→ #0])).ex, by intro v; simp [hp.df.iff, hpf.df.iff, HSemiformula.val_sigma]  ⟩
    ⟨(pf.negSigma ⋎ (p.pi.rew [→ #0])).all, by intro v; simp [hp.df.iff, hpf.df.iff, ←imp_iff_not_or, hp.proper.iff']⟩
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 → Prop
f : (Fin k → M) → M
hf : DefinableFunction L (𝚺, m + 1) f
Γ : SigmaPiDelta
hP : DefinablePred L (Γ, m + 1) P
⊢ Definable L (Γ, m + 1) fun v => P (f v)
no goals