Theorem: Arith.Model.DefinableFunction.const (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
k : ℕ
c : M
⊢ Hierarchy 𝚺 0 (“#0 = &c”)
Tactic:
simp
State after:
no goals

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
k : ℕ
c : M
⊢ DefinedWithParam (fun v => v 0 = (fun x => c) fun x => v x.succ) (HSemiformula.mkSigma (“#0 = &c”) ⋯)
Tactic:
intro v
State after:
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
k : ℕ
c : M
v : Fin (k + 1) → M
⊢ (fun v => v 0 = (fun x => c) fun x => v x.succ) v ↔
    (Semiformula.Evalm M v id) (HSemiformula.mkSigma (“#0 = &c”) ⋯).val

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
k : ℕ
c : M
v : Fin (k + 1) → M
⊢ (fun v => v 0 = (fun x => c) fun x => v x.succ) v ↔
    (Semiformula.Evalm M v id) (HSemiformula.mkSigma (“#0 = &c”) ⋯).val
Tactic:
simp
State after:
no goals