Theorem: Arith.Model.DefinableFunction₂.comp (Arithmetization/Definability/Hierarchy.lean)

State before:
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 : ℕ
Γ : SigmaPiDelta
k : ℕ
f : M → M → M
g₁ g₂ : (Fin k → M) → M
hf : DefinableFunction₂ L (Γ, m + 1) f
hg₁ : DefinableFunction L (𝚺, m + 1) g₁
hg₂ : DefinableFunction L (𝚺, m + 1) g₂
⊢ DefinableFunction L (Γ, m + 1) fun v => f (g₁ v) (g₂ v)
Tactic:
simpa using Definable.comp₃ (P := Function.Graph₂ f) (DefinableFunction.var 0) (hg₁.retraction Fin.succ) (hg₂.retraction Fin.succ) hf
State after:
no goals