internlm/Lean-Github huggingface dataset:
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 (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)
Tactic:
exact ⟨(pf.negSigma ⋎ (p.rew [→ #0])).all, by intro v; simp [hp.df.iff, hpf.df.iff, ←imp_iff_not_or]⟩
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 (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
⊢ DefinedWithParam (fun v => P (f v)) (pf.negSigma ⋎ HSemiformula.rew [→ #0] p).all
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 (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
v : Fin k → M
⊢ (fun v => P (f v)) v ↔ (Semiformula.Evalm M v id) (pf.negSigma ⋎ HSemiformula.rew [→ #0] p).all.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 (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
v : Fin k → M
⊢ (fun v => P (f v)) v ↔ (Semiformula.Evalm M v id) (pf.negSigma ⋎ HSemiformula.rew [→ #0] p).all.val
Tactic:
simp [hp.df.iff, hpf.df.iff, ←imp_iff_not_or]
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 (𝚫, m + 1) P
⊢ Definable L (𝚫, 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 (𝚫, m + 1)
hp : DefinedWithParam (fun v => P (v 0)) p
⊢ Definable L (𝚫, 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 (𝚫, m + 1)
hp : DefinedWithParam (fun v => P (v 0)) p
⊢ Definable L (𝚫, 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 (𝚫, 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 (𝚫, 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 (𝚫, 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 (𝚫, m + 1) fun v => P (f v)
Tactic:
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
Γ : SigmaPiDelta
p : HSemiformula L M 1 (𝚫, 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.sigma).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 (𝚫, 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.sigma).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 (𝚫, 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.sigma).ex.val
Tactic:
simp [hp.df.iff, hpf.df.iff, HSemiformula.val_sigma]
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 (𝚫, 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.negSigma ⋎ HSemiformula.rew [→ #0] p.pi).all
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 (𝚫, 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.negSigma ⋎ HSemiformula.rew [→ #0] p.pi).all.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 (𝚫, 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.negSigma ⋎ HSemiformula.rew [→ #0] p.pi).all.val
Tactic:
simp [hp.df.iff, hpf.df.iff, ←imp_iff_not_or, hp.proper.iff']
Proof state after:
no goals
Theorem: Arith.Model.Definable.comp₂
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 → M → Prop
f g : (Fin k → M) → M
hf : DefinableFunction L (𝚺, m + 1) f
hg : DefinableFunction L (𝚺, m + 1) g
Γ : SigmaPiDelta
hP : DefinableRel L (𝚺, m + 1) P
⊢ Definable L (𝚺, m + 1) fun v => P (f v) (g 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 → M → Prop
f g : (Fin k → M) → M
hf : DefinableFunction L (𝚺, m + 1) f
hg : DefinableFunction L (𝚺, m + 1) g
Γ : SigmaPiDelta
p : HSemiformula L M 2 (𝚺, m + 1)
hp : DefinedWithParam (fun v => P (v 0) (v 1)) p
⊢ Definable L (𝚺, m + 1) fun v => P (f v) (g v)
Theorem: Arith.Model.Definable.comp₂
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 → M → Prop
f g : (Fin k → M) → M
hf : DefinableFunction L (𝚺, m + 1) f
hg : DefinableFunction L (𝚺, m + 1) g
Γ : SigmaPiDelta
p : HSemiformula L M 2 (𝚺, m + 1)
hp : DefinedWithParam (fun v => P (v 0) (v 1)) p
⊢ Definable L (𝚺, m + 1) fun v => P (f v) (g 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 → M → Prop
f g : (Fin k → M) → M
hg : DefinableFunction L (𝚺, m + 1) g
Γ : SigmaPiDelta
p : HSemiformula L M 2 (𝚺, m + 1)
hp : DefinedWithParam (fun v => P (v 0) (v 1)) 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) (g v)
Theorem: Arith.Model.Definable.comp₂
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 → M → Prop
f g : (Fin k → M) → M
hg : DefinableFunction L (𝚺, m + 1) g
Γ : SigmaPiDelta
p : HSemiformula L M 2 (𝚺, m + 1)
hp : DefinedWithParam (fun v => P (v 0) (v 1)) 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) (g v)
Tactic:
rcases hg with ⟨pg, hpg⟩
Proof state after:
case mk.intro.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 → M → Prop
f g : (Fin k → M) → M
Γ : SigmaPiDelta
p : HSemiformula L M 2 (𝚺, m + 1)
hp : DefinedWithParam (fun v => P (v 0) (v 1)) p
pf : HSemiformula L M (k + 1) (𝚺, m + 1)
hpf : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) pf
pg : HSemiformula L M (k + 1) (𝚺, m + 1)
hpg : DefinedWithParam (fun v => v 0 = g fun x => v x.succ) pg
⊢ Definable L (𝚺, m + 1) fun v => P (f v) (g v)
Theorem: Arith.Model.Definable.comp₂
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 → M → Prop
f g : (Fin k → M) → M
Γ : SigmaPiDelta
p : HSemiformula L M 2 (𝚺, m + 1)
hp : DefinedWithParam (fun v => P (v 0) (v 1)) p
pf : HSemiformula L M (k + 1) (𝚺, m + 1)
hpf : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) pf
pg : HSemiformula L M (k + 1) (𝚺, m + 1)
hpg : DefinedWithParam (fun v => v 0 = g fun x => v x.succ) pg
⊢ DefinedWithParam (fun v => P (f v) (g v))
    (HSemiformula.rew (Rew.substs (#0 :> fun x => #x.succ.succ)) pf ⋏
          HSemiformula.rew (Rew.substs (#1 :> fun x => #x.succ.succ)) pg ⋏ HSemiformula.rew [→ #0, #1] p).ex.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 → M → Prop
f g : (Fin k → M) → M
Γ : SigmaPiDelta
p : HSemiformula L M 2 (𝚺, m + 1)
hp : DefinedWithParam (fun v => P (v 0) (v 1)) p
pf : HSemiformula L M (k + 1) (𝚺, m + 1)
hpf : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) pf
pg : HSemiformula L M (k + 1) (𝚺, m + 1)
hpg : DefinedWithParam (fun v => v 0 = g fun x => v x.succ) pg
v : Fin k → M
⊢ (fun v => P (f v) (g v)) v ↔
    (Semiformula.Evalm M v id)
      (HSemiformula.rew (Rew.substs (#0 :> fun x => #x.succ.succ)) pf ⋏
              HSemiformula.rew (Rew.substs (#1 :> fun x => #x.succ.succ)) pg ⋏ HSemiformula.rew [→ #0, #1] p).ex.ex.val
Theorem: Arith.Model.Definable.comp₂
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 → M → Prop
f g : (Fin k → M) → M
Γ : SigmaPiDelta
p : HSemiformula L M 2 (𝚺, m + 1)
hp : DefinedWithParam (fun v => P (v 0) (v 1)) p
pf : HSemiformula L M (k + 1) (𝚺, m + 1)
hpf : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) pf
pg : HSemiformula L M (k + 1) (𝚺, m + 1)
hpg : DefinedWithParam (fun v => v 0 = g fun x => v x.succ) pg
v : Fin k → M
⊢ (fun v => P (f v) (g v)) v ↔
    (Semiformula.Evalm M v id)
      (HSemiformula.rew (Rew.substs (#0 :> fun x => #x.succ.succ)) pf ⋏
              HSemiformula.rew (Rew.substs (#1 :> fun x => #x.succ.succ)) pg ⋏ HSemiformula.rew [→ #0, #1] p).ex.ex.val
Tactic:
simp [hp.df.iff, hpf.df.iff, hpg.df.iff]
Proof state after:
no goals
Theorem: Arith.Model.Definable.comp₂
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 → M → Prop
f g : (Fin k → M) → M
hf : DefinableFunction L (𝚺, m + 1) f
hg : DefinableFunction L (𝚺, m + 1) g
Γ : SigmaPiDelta
hP : DefinableRel L (𝚷, m + 1) P
⊢ Definable L (𝚷, m + 1) fun v => P (f v) (g 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 → M → Prop
f g : (Fin k → M) → M
hf : DefinableFunction L (𝚺, m + 1) f
hg : DefinableFunction L (𝚺, m + 1) g
Γ : SigmaPiDelta
p : HSemiformula L M 2 (𝚷, m + 1)
hp : DefinedWithParam (fun v => P (v 0) (v 1)) p
⊢ Definable L (𝚷, m + 1) fun v => P (f v) (g v)
Theorem: Arith.Model.Definable.comp₂
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 → M → Prop
f g : (Fin k → M) → M
hf : DefinableFunction L (𝚺, m + 1) f
hg : DefinableFunction L (𝚺, m + 1) g
Γ : SigmaPiDelta
p : HSemiformula L M 2 (𝚷, m + 1)
hp : DefinedWithParam (fun v => P (v 0) (v 1)) p
⊢ Definable L (𝚷, m + 1) fun v => P (f v) (g 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 → M → Prop
f g : (Fin k → M) → M
hg : DefinableFunction L (𝚺, m + 1) g
Γ : SigmaPiDelta
p : HSemiformula L M 2 (𝚷, m + 1)
hp : DefinedWithParam (fun v => P (v 0) (v 1)) 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) (g v)
Theorem: Arith.Model.Definable.comp₂
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 → M → Prop
f g : (Fin k → M) → M
hg : DefinableFunction L (𝚺, m + 1) g
Γ : SigmaPiDelta
p : HSemiformula L M 2 (𝚷, m + 1)
hp : DefinedWithParam (fun v => P (v 0) (v 1)) 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) (g v)
Tactic:
rcases hg with ⟨pg, hpg⟩
Proof state after:
case mk.intro.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 → M → Prop
f g : (Fin k → M) → M
Γ : SigmaPiDelta
p : HSemiformula L M 2 (𝚷, m + 1)
hp : DefinedWithParam (fun v => P (v 0) (v 1)) p
pf : HSemiformula L M (k + 1) (𝚺, m + 1)
hpf : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) pf
pg : HSemiformula L M (k + 1) (𝚺, m + 1)
hpg : DefinedWithParam (fun v => v 0 = g fun x => v x.succ) pg
⊢ Definable L (𝚷, m + 1) fun v => P (f v) (g v)
Theorem: Arith.Model.Definable.comp₂
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 → M → Prop
f g : (Fin k → M) → M
Γ : SigmaPiDelta
p : HSemiformula L M 2 (𝚷, m + 1)
hp : DefinedWithParam (fun v => P (v 0) (v 1)) p
pf : HSemiformula L M (k + 1) (𝚺, m + 1)
hpf : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) pf
pg : HSemiformula L M (k + 1) (𝚺, m + 1)
hpg : DefinedWithParam (fun v => v 0 = g fun x => v x.succ) pg
⊢ DefinedWithParam (fun v => P (f v) (g v))
    ((HSemiformula.rew (Rew.substs (#0 :> fun x => #x.succ.succ)) pf).negSigma ⋎
          (HSemiformula.rew (Rew.substs (#1 :> fun x => #x.succ.succ)) pg).negSigma ⋎
            HSemiformula.rew [→ #0, #1] p).all.all
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 → M → Prop
f g : (Fin k → M) → M
Γ : SigmaPiDelta
p : HSemiformula L M 2 (𝚷, m + 1)
hp : DefinedWithParam (fun v => P (v 0) (v 1)) p
pf : HSemiformula L M (k + 1) (𝚺, m + 1)
hpf : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) pf
pg : HSemiformula L M (k + 1) (𝚺, m + 1)
hpg : DefinedWithParam (fun v => v 0 = g fun x => v x.succ) pg
v : Fin k → M
⊢ (fun v => P (f v) (g v)) v ↔
    (Semiformula.Evalm M v id)
      ((HSemiformula.rew (Rew.substs (#0 :> fun x => #x.succ.succ)) pf).negSigma ⋎
              (HSemiformula.rew (Rew.substs (#1 :> fun x => #x.succ.succ)) pg).negSigma ⋎
                HSemiformula.rew [→ #0, #1] p).all.all.val