internlm/Lean-Github huggingface dataset:
Theorem: Arith.Model.Definable.imp
Proof 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
m : ℕ
Γ : SigmaPiDelta
h₂ : Definable L (𝚷, m) Q
p₁ : HSemiformula L M k (𝚷.alt, m)
h₁ : DefinedWithParam P p₁
⊢ Definable L (𝚷, m) fun v => P v → Q v
Tactic:
rcases h₂ with ⟨p₂, h₂⟩
Proof state after:
case mk.intro.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
m : ℕ
Γ : SigmaPiDelta
p₁ : HSemiformula L M k (𝚷.alt, m)
h₁ : DefinedWithParam P p₁
p₂ : HSemiformula L M k (𝚷, m)
h₂ : DefinedWithParam Q p₂
⊢ Definable L (𝚷, m) fun v => P v → Q v
Theorem: Arith.Model.Definable.imp
Proof state before:
case mk.intro.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
m : ℕ
Γ : SigmaPiDelta
p₁ : HSemiformula L M k (𝚷.alt, m)
h₁ : DefinedWithParam P p₁
p₂ : HSemiformula L M k (𝚷, m)
h₂ : DefinedWithParam Q p₂
⊢ Definable L (𝚷, m) fun v => P v → Q v
Tactic:
exact ⟨p₁.negSigma.or p₂, (h₁.negSigma.or h₂).of_iff (fun x ↦ by simp [imp_iff_not_or])⟩
Proof state after:
no goals
Theorem: Arith.Model.Definable.imp
Proof 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
p₁ : HSemiformula L M k (𝚷.alt, m)
h₁ : DefinedWithParam P p₁
p₂ : HSemiformula L M k (𝚷, m)
h₂ : DefinedWithParam Q p₂
x : Fin k → M
⊢ P x → Q x ↔ ¬P x ∨ Q x
Tactic:
simp [imp_iff_not_or]
Proof state after:
no goals
Theorem: Arith.Model.Definable.imp
Proof 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
h₁ : Definable L (𝚫.alt, m) P
h₂ : Definable L (𝚫, m) Q
⊢ Definable L (𝚫, m) fun v => P v → Q v
Tactic:
rcases h₁ with ⟨p₁, h₁⟩
Proof 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
m : ℕ
Γ : SigmaPiDelta
h₂ : Definable L (𝚫, m) Q
p₁ : HSemiformula L M k (𝚫.alt, m)
h₁ : DefinedWithParam P p₁
⊢ Definable L (𝚫, m) fun v => P v → Q v
Theorem: Arith.Model.Definable.imp
Proof 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
m : ℕ
Γ : SigmaPiDelta
h₂ : Definable L (𝚫, m) Q
p₁ : HSemiformula L M k (𝚫.alt, m)
h₁ : DefinedWithParam P p₁
⊢ Definable L (𝚫, m) fun v => P v → Q v
Tactic:
rcases h₂ with ⟨p₂, h₂⟩
Proof state after:
case mk.intro.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
m : ℕ
Γ : SigmaPiDelta
p₁ : HSemiformula L M k (𝚫.alt, m)
h₁ : DefinedWithParam P p₁
p₂ : HSemiformula L M k (𝚫, m)
h₂ : DefinedWithParam Q p₂
⊢ Definable L (𝚫, m) fun v => P v → Q v
Theorem: Arith.Model.Definable.imp
Proof state before:
case mk.intro.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
m : ℕ
Γ : SigmaPiDelta
p₁ : HSemiformula L M k (𝚫.alt, m)
h₁ : DefinedWithParam P p₁
p₂ : HSemiformula L M k (𝚫, m)
h₂ : DefinedWithParam Q p₂
⊢ Definable L (𝚫, m) fun v => P v → Q v
Tactic:
exact ⟨p₁ ⟶ p₂, h₁.imp h₂⟩
Proof state after:
no goals
Theorem: Arith.Model.Definable.iff
Proof 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
h₁ : Definable L (𝚫, m) P
h₂ : Definable L (𝚫, m) Q
Γ : SigmaPiDelta
⊢ Definable L (𝚫, m) fun v => P v ↔ Q v
Tactic:
rcases h₁ with ⟨p, hp⟩
Proof 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
m : ℕ
h₂ : Definable L (𝚫, m) Q
Γ : SigmaPiDelta
p : HSemiformula L M k (𝚫, m)
hp : DefinedWithParam P p
⊢ Definable L (𝚫, m) fun v => P v ↔ Q v
Theorem: Arith.Model.Definable.iff
Proof 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
m : ℕ
h₂ : Definable L (𝚫, m) Q
Γ : SigmaPiDelta
p : HSemiformula L M k (𝚫, m)
hp : DefinedWithParam P p
⊢ Definable L (𝚫, m) fun v => P v ↔ Q v
Tactic:
rcases h₂ with ⟨q, hq⟩
Proof state after:
case mk.intro.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
m : ℕ
Γ : SigmaPiDelta
p : HSemiformula L M k (𝚫, m)
hp : DefinedWithParam P p
q : HSemiformula L M k (𝚫, m)
hq : DefinedWithParam Q q
⊢ Definable L (𝚫, m) fun v => P v ↔ Q v
Theorem: Arith.Model.Definable.iff
Proof state before:
case mk.intro.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
m : ℕ
Γ : SigmaPiDelta
p : HSemiformula L M k (𝚫, m)
hp : DefinedWithParam P p
q : HSemiformula L M k (𝚫, m)
hq : DefinedWithParam Q q
⊢ Definable L (𝚫, m) fun v => P v ↔ Q v
Tactic:
exact ⟨p ⟷ q, hp.iff hq⟩
Proof state after:
no goals
Theorem: Arith.Model.Definable.all
Proof 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 (𝚷, s + 1) fun w => P (fun x => w x.succ) (w 0)
⊢ Definable L (𝚷, s + 1) fun v => ∀ (x : M), P v x
Tactic:
rcases h with ⟨p, hp⟩
Proof 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 (𝚷, s + 1)
hp : DefinedWithParam (fun w => P (fun x => w x.succ) (w 0)) p
⊢ Definable L (𝚷, s + 1) fun v => ∀ (x : M), P v x
Theorem: Arith.Model.Definable.all
Proof 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 (𝚷, s + 1)
hp : DefinedWithParam (fun w => P (fun x => w x.succ) (w 0)) p
⊢ Definable L (𝚷, s + 1) fun v => ∀ (x : M), P v x
Tactic:
exact ⟨.mkPi (∀' p.val) (by simp), by intro v; simp [hp.df.iff]⟩
Proof state after:
no goals
Theorem: Arith.Model.Definable.all
Proof 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 (𝚷, s + 1)
hp : DefinedWithParam (fun w => P (fun x => w x.succ) (w 0)) p
⊢ Hierarchy 𝚷 (s + 1) (∀' p.val)
Tactic:
simp
Proof state after:
no goals
Theorem: Arith.Model.Definable.all
Proof 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 (𝚷, s + 1)
hp : DefinedWithParam (fun w => P (fun x => w x.succ) (w 0)) p
⊢ DefinedWithParam (fun v => ∀ (x : M), P v x) (HSemiformula.mkPi (∀' p.val) ⋯)
Tactic:
intro v
Proof 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 (𝚷, s + 1)
hp : DefinedWithParam (fun w => P (fun x => w x.succ) (w 0)) p
v : Fin k → M
⊢ (fun v => ∀ (x : M), P v x) v ↔ (Semiformula.Evalm M v id) (HSemiformula.mkPi (∀' p.val) ⋯).val
Theorem: Arith.Model.Definable.all
Proof 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 (𝚷, s + 1)
hp : DefinedWithParam (fun w => P (fun x => w x.succ) (w 0)) p
v : Fin k → M
⊢ (fun v => ∀ (x : M), P v x) v ↔ (Semiformula.Evalm M v id) (HSemiformula.mkPi (∀' p.val) ⋯).val
Tactic:
simp [hp.df.iff]
Proof state after:
no goals
Theorem: Arith.Model.Definable.ex
Proof 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⟩
Proof 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
Theorem: Arith.Model.Definable.ex
Proof 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]⟩
Proof state after:
no goals
Theorem: Arith.Model.Definable.ex
Proof 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
Proof state after:
no goals
Theorem: Arith.Model.Definable.ex
Proof 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
Proof 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
Theorem: Arith.Model.Definable.ex
Proof 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]
Proof state after:
no goals
Theorem: Arith.Model.Definable.comp₁
Proof 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 k : ℕ
P : M → Prop
f : (Fin k → M) → M
hf : DefinableFunction L (𝚺, m + 1) f
Γ : SigmaPiDelta
hP : DefinablePred L (Γ, m + 1) P
⊢ Definable L (Γ, m + 1) fun v => P (f v)
Tactic:
match Γ with
| 𝚺 =>
  rcases hP with ⟨p, hp⟩; rcases hf with ⟨pf, hpf⟩
  exact ⟨(pf ⋏ (p.rew [→ #0])).ex, by intro v; simp [hp.df.iff, hpf.df.iff]⟩
| 𝚷 =>
  rcases hP with ⟨p, hp⟩; rcases hf with ⟨pf, hpf⟩
  exact ⟨(pf.negSigma ⋎ (p.rew [→ #0])).all, by intro v; simp [hp.df.iff, hpf.df.iff, ←imp_iff_not_or]⟩
| 𝚫 =>
  rcases hP with ⟨p, hp⟩; rcases hf with ⟨pf, hpf⟩
  exact of_sigma_of_pi
    ⟨(pf ⋏ (p.sigma.rew [→ #0])).ex, by intro v; simp [hp.df.iff, hpf.df.iff, HSemiformula.val_sigma]  ⟩
    ⟨(pf.negSigma ⋎ (p.pi.rew [→ #0])).all, by intro v; simp [hp.df.iff, hpf.df.iff, ←imp_iff_not_or, hp.proper.iff']⟩
Proof state after:
no goals
Theorem: Arith.Model.Definable.comp₁
Proof 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 k : ℕ
P : M → Prop
f : (Fin k → M) → M
hf : DefinableFunction L (𝚺, m + 1) f
Γ : SigmaPiDelta
hP : DefinablePred L (𝚺, m + 1) P
⊢ Definable L (𝚺, m + 1) fun v => P (f v)
Tactic:
rcases hP with ⟨p, hp⟩
Proof 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
m k : ℕ
P : M → Prop
f : (Fin k → M) → M
hf : DefinableFunction L (𝚺, m + 1) f
Γ : SigmaPiDelta
p : HSemiformula L M 1 (𝚺, m + 1)
hp : DefinedWithParam (fun v => P (v 0)) p
⊢ Definable L (𝚺, m + 1) fun v => P (f v)
Theorem: Arith.Model.Definable.comp₁
Proof 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
m k : ℕ
P : M → Prop
f : (Fin k → M) → M
hf : DefinableFunction L (𝚺, m + 1) f
Γ : SigmaPiDelta
p : HSemiformula L M 1 (𝚺, m + 1)
hp : DefinedWithParam (fun v => P (v 0)) p
⊢ Definable L (𝚺, m + 1) fun v => P (f v)
Tactic:
rcases hf with ⟨pf, hpf⟩
Proof state after:
case mk.intro.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
m k : ℕ
P : M → Prop
f : (Fin k → M) → M
Γ : SigmaPiDelta
p : HSemiformula L M 1 (𝚺, m + 1)
hp : DefinedWithParam (fun v => P (v 0)) p
pf : HSemiformula L M (k + 1) (𝚺, m + 1)
hpf : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) pf
⊢ Definable L (𝚺, m + 1) fun v => P (f v)
Theorem: Arith.Model.Definable.comp₁
Proof state before:
case mk.intro.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
m k : ℕ
P : M → Prop
f : (Fin k → M) → M
Γ : SigmaPiDelta
p : HSemiformula L M 1 (𝚺, m + 1)
hp : DefinedWithParam (fun v => P (v 0)) p
pf : HSemiformula L M (k + 1) (𝚺, m + 1)
hpf : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) pf
⊢ Definable L (𝚺, m + 1) fun v => P (f v)
Tactic:
exact ⟨(pf ⋏ (p.rew [→ #0])).ex, by intro v; simp [hp.df.iff, hpf.df.iff]⟩
Proof state after:
no goals
Theorem: Arith.Model.Definable.comp₁
Proof 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 k : ℕ
P : M → Prop
f : (Fin k → M) → M
Γ : SigmaPiDelta
p : HSemiformula L M 1 (𝚺, m + 1)
hp : DefinedWithParam (fun v => P (v 0)) p
pf : HSemiformula L M (k + 1) (𝚺, m + 1)
hpf : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) pf
⊢ DefinedWithParam (fun v => P (f v)) (pf ⋏ HSemiformula.rew [→ #0] p).ex
Tactic:
intro v
Proof 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
m k : ℕ
P : M → Prop
f : (Fin k → M) → M
Γ : SigmaPiDelta
p : HSemiformula L M 1 (𝚺, m + 1)
hp : DefinedWithParam (fun v => P (v 0)) p
pf : HSemiformula L M (k + 1) (𝚺, m + 1)
hpf : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) pf
v : Fin k → M
⊢ (fun v => P (f v)) v ↔ (Semiformula.Evalm M v id) (pf ⋏ HSemiformula.rew [→ #0] p).ex.val
Theorem: Arith.Model.Definable.comp₁
Proof 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 k : ℕ
P : M → Prop
f : (Fin k → M) → M
Γ : SigmaPiDelta
p : HSemiformula L M 1 (𝚺, m + 1)
hp : DefinedWithParam (fun v => P (v 0)) p
pf : HSemiformula L M (k + 1) (𝚺, m + 1)
hpf : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) pf
v : Fin k → M
⊢ (fun v => P (f v)) v ↔ (Semiformula.Evalm M v id) (pf ⋏ HSemiformula.rew [→ #0] p).ex.val
Tactic:
simp [hp.df.iff, hpf.df.iff]
Proof state after:
no goals
Theorem: Arith.Model.Definable.comp₁
Proof 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 k : ℕ
P : M → Prop
f : (Fin k → M) → M
hf : DefinableFunction L (𝚺, m + 1) f
Γ : SigmaPiDelta
hP : DefinablePred L (𝚷, m + 1) P
⊢ Definable L (𝚷, m + 1) fun v => P (f v)
Tactic:
rcases hP with ⟨p, hp⟩
Proof 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
m k : ℕ
P : M → Prop
f : (Fin k → M) → M
hf : DefinableFunction L (𝚺, m + 1) f
Γ : SigmaPiDelta
p : HSemiformula L M 1 (𝚷, m + 1)
hp : DefinedWithParam (fun v => P (v 0)) p
⊢ Definable L (𝚷, m + 1) fun v => P (f v)
Theorem: Arith.Model.Definable.comp₁
Proof 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
m k : ℕ
P : M → Prop
f : (Fin k → M) → M
hf : DefinableFunction L (𝚺, m + 1) f
Γ : SigmaPiDelta
p : HSemiformula L M 1 (𝚷, m + 1)
hp : DefinedWithParam (fun v => P (v 0)) p
⊢ Definable L (𝚷, m + 1) fun v => P (f v)
Tactic:
rcases hf with ⟨pf, hpf⟩
Proof state after:
case mk.intro.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
m k : ℕ
P : M → Prop
f : (Fin k → M) → M
Γ : SigmaPiDelta
p : HSemiformula L M 1 (𝚷, m + 1)
hp : DefinedWithParam (fun v => P (v 0)) p
pf : HSemiformula L M (k + 1) (𝚺, m + 1)
hpf : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) pf
⊢ Definable L (𝚷, m + 1) fun v => P (f v)