Theorem: LO.FirstOrder.Arith.Model.DefinableBoundedFunction.of_zero (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
k : ℕ
Γ' Γ : SigmaPiDelta
f : (Fin k → M) → M
h : DefinableBoundedFunction L (Γ', 0) f
⊢ DefinableBoundedFunction L (Γ, 0) f
Tactic:
rcases h with ⟨hb, h⟩
State after:
case intro
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
k : ℕ
Γ' Γ : SigmaPiDelta
f : (Fin k → M) → M
hb : Bounded L f
h : DefinableFunction L (Γ', 0) f
⊢ DefinableBoundedFunction L (Γ, 0) f

State before:
case intro
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
k : ℕ
Γ' Γ : SigmaPiDelta
f : (Fin k → M) → M
hb : Bounded L f
h : DefinableFunction L (Γ', 0) f
⊢ DefinableBoundedFunction L (Γ, 0) f
Tactic:
exact ⟨hb, .of_zero h _⟩
State after:
no goals