Theorem: Arith.Model.Definable.ex (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
s : ℕ
P : (Fin k → M) → M → Prop
h : Definable L (𝚺, s + 1) fun w => P (fun x => w x.succ) (w 0)
⊢ Definable L (𝚺, s + 1) fun v => ∃ x, P v x
Tactic:
rcases h with ⟨p, hp⟩
State after:
case mk.intro
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
s : ℕ
P : (Fin k → M) → M → Prop
p : HSemiformula L M k.succ (𝚺, s + 1)
hp : DefinedWithParam (fun w => P (fun x => w x.succ) (w 0)) p
⊢ Definable L (𝚺, s + 1) fun v => ∃ x, P v x

State before:
case mk.intro
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
s : ℕ
P : (Fin k → M) → M → Prop
p : HSemiformula L M k.succ (𝚺, s + 1)
hp : DefinedWithParam (fun w => P (fun x => w x.succ) (w 0)) p
⊢ Definable L (𝚺, s + 1) fun v => ∃ x, P v x
Tactic:
exact ⟨.mkSigma (∃' p.val) (by simp), by intro v; simp [hp.df.iff]⟩
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
s : ℕ
P : (Fin k → M) → M → Prop
p : HSemiformula L M k.succ (𝚺, s + 1)
hp : DefinedWithParam (fun w => P (fun x => w x.succ) (w 0)) p
⊢ Hierarchy 𝚺 (s + 1) (∃' p.val)
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
s : ℕ
P : (Fin k → M) → M → Prop
p : HSemiformula L M k.succ (𝚺, s + 1)
hp : DefinedWithParam (fun w => P (fun x => w x.succ) (w 0)) p
⊢ DefinedWithParam (fun v => ∃ x, P v x) (HSemiformula.mkSigma (∃' p.val) ⋯)
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
s : ℕ
P : (Fin k → M) → M → Prop
p : HSemiformula L M k.succ (𝚺, s + 1)
hp : DefinedWithParam (fun w => P (fun x => w x.succ) (w 0)) p
v : Fin k → M
⊢ (fun v => ∃ x, P v x) v ↔ (Semiformula.Evalm M v id) (HSemiformula.mkSigma (∃' p.val) ⋯).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
s : ℕ
P : (Fin k → M) → M → Prop
p : HSemiformula L M k.succ (𝚺, s + 1)
hp : DefinedWithParam (fun w => P (fun x => w x.succ) (w 0)) p
v : Fin k → M
⊢ (fun v => ∃ x, P v x) v ↔ (Semiformula.Evalm M v id) (HSemiformula.mkSigma (∃' p.val) ⋯).val
Tactic:
simp [hp.df.iff]
State after:
no goals