Theorem: LO.FirstOrder.Arith.Model.Bounded₃.comp (Arithmetization/Definability/Definability.lean)

State before:
L : Language
inst✝⁹ : L.ORing
ξ : Type u_1
n : ℕ
M : Type u_2
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
f : M → M → M → M
k : ℕ
g₁ g₂ g₃ : (Fin k → M) → M
hf : Bounded₃ L f
hg₁ : Bounded L g₁
hg₂ : Bounded L g₂
hg₃ : Bounded L g₃
i : Fin (Nat.succ 0).succ.succ
⊢ Bounded L (![g₁, g₂, g₃] i)
Tactic:
cases' i using Fin.cases with i <;> simp [*]
State after:
case succ
L : Language
inst✝⁹ : L.ORing
ξ : Type u_1
n : ℕ
M : Type u_2
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
f : M → M → M → M
k : ℕ
g₁ g₂ g₃ : (Fin k → M) → M
hf : Bounded₃ L f
hg₁ : Bounded L g₁
hg₂ : Bounded L g₂
hg₃ : Bounded L g₃
i : Fin 2
⊢ Bounded L (![g₂, g₃] i)

State before:
case succ
L : Language
inst✝⁹ : L.ORing
ξ : Type u_1
n : ℕ
M : Type u_2
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
f : M → M → M → M
k : ℕ
g₁ g₂ g₃ : (Fin k → M) → M
hf : Bounded₃ L f
hg₁ : Bounded L g₁
hg₂ : Bounded L g₂
hg₃ : Bounded L g₃
i : Fin 2
⊢ Bounded L (![g₂, g₃] i)
Tactic:
cases' i using Fin.cases with i <;> simp [*]
State after:
no goals