internlm/Lean-Github huggingface dataset:
Theorem: Arith.Model.DefinableFunction₂.comp
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
k : ℕ
f : M → M → M
g₁ g₂ : (Fin k → M) → M
hf : DefinableFunction₂ L (Γ, m + 1) f
hg₁ : DefinableFunction L (𝚺, m + 1) g₁
hg₂ : DefinableFunction L (𝚺, m + 1) g₂
⊢ DefinableFunction L (Γ, m + 1) fun v => f (g₁ v) (g₂ v)
Tactic:
simpa using Definable.comp₃ (P := Function.Graph₂ f) (DefinableFunction.var 0) (hg₁.retraction Fin.succ) (hg₂.retraction Fin.succ) hf
Proof state after:
no goals
Theorem: Arith.Model.DefinableFunction₃.comp
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
k : ℕ
f : M → M → M → M
g₁ g₂ g₃ : (Fin k → M) → M
hf : DefinableFunction₃ L (Γ, m + 1) f
hg₁ : DefinableFunction L (𝚺, m + 1) g₁
hg₂ : DefinableFunction L (𝚺, m + 1) g₂
hg₃ : DefinableFunction L (𝚺, m + 1) g₃
⊢ DefinableFunction L (Γ, m + 1) fun v => f (g₁ v) (g₂ v) (g₃ v)
Tactic:
simpa using Definable.comp₄ (P := Function.Graph₃ f) (DefinableFunction.var 0)
  (hg₁.retraction Fin.succ) (hg₂.retraction Fin.succ) (hg₃.retraction Fin.succ) hf
Proof state after:
no goals
Theorem: Arith.Model.Definable.ball_lt
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
P : (Fin k → M) → M → Prop
f : (Fin k → M) → M
hf : DefinableFunction L (𝚺, m + 1) f
h : Definable L (Γ, m + 1) fun w => P (fun x => w x.succ) (w 0)
⊢ Definable L (Γ, m + 1) fun v => ∀ x < f v, P v x
Tactic:
rcases hf with ⟨bf, hbf⟩
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
P : (Fin k → M) → M → Prop
f : (Fin k → M) → M
h : Definable L (Γ, m + 1) fun w => P (fun x => w x.succ) (w 0)
bf : HSemiformula L M (k + 1) (𝚺, m + 1)
hbf : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) bf
⊢ Definable L (Γ, m + 1) fun v => ∀ x < f v, P v x
Theorem: Arith.Model.Definable.ball_lt
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
P : (Fin k → M) → M → Prop
f : (Fin k → M) → M
h : Definable L (Γ, m + 1) fun w => P (fun x => w x.succ) (w 0)
bf : HSemiformula L M (k + 1) (𝚺, m + 1)
hbf : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) bf
⊢ Definable L (Γ, m + 1) fun v => ∀ x < f v, P v x
Tactic:
rcases h with ⟨p, hp⟩
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 : (Fin k → M) → M → Prop
f : (Fin k → M) → M
bf : HSemiformula L M (k + 1) (𝚺, m + 1)
hbf : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) bf
p : HSemiformula L M k.succ (Γ, m + 1)
hp : DefinedWithParam (fun w => P (fun x => w x.succ) (w 0)) p
⊢ Definable L (Γ, m + 1) fun v => ∀ x < f v, P v x
Theorem: Arith.Model.Definable.ball_lt
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
P : (Fin k → M) → M → Prop
f : (Fin k → M) → M
bf : HSemiformula L M (k + 1) (𝚺, m + 1)
hbf : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) bf
p : HSemiformula L M k.succ (𝚺, m + 1)
hp : DefinedWithParam (fun w => P (fun x => w x.succ) (w 0)) p
⊢ Hierarchy 𝚺 (m + 1) (“∃' (!bf.val ∧ (∀[#0 < #1] !((Rew.substs (#0 :> fun x => #x.succ.succ)).hom p.val)))”)
Tactic:
simp
Proof state after:
no goals
Theorem: Arith.Model.Definable.ball_lt
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
P : (Fin k → M) → M → Prop
f : (Fin k → M) → M
bf : HSemiformula L M (k + 1) (𝚺, m + 1)
hbf : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) bf
p : HSemiformula L M k.succ (𝚺, m + 1)
hp : DefinedWithParam (fun w => P (fun x => w x.succ) (w 0)) p
⊢ DefinedWithParam (fun v => ∀ x < f v, P v x)
    (HSemiformula.mkSigma (“∃' (!bf.val ∧ (∀[#0 < #1] !((Rew.substs (#0 :> fun x => #x.succ.succ)).hom 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
m : ℕ
Γ : SigmaPiDelta
P : (Fin k → M) → M → Prop
f : (Fin k → M) → M
bf : HSemiformula L M (k + 1) (𝚺, m + 1)
hbf : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) bf
p : HSemiformula L M k.succ (𝚺, m + 1)
hp : DefinedWithParam (fun w => P (fun x => w x.succ) (w 0)) p
v : Fin k → M
⊢ (fun v => ∀ x < f v, P v x) v ↔
    (Semiformula.Evalm M v id)
      (HSemiformula.mkSigma (“∃' (!bf.val ∧ (∀[#0 < #1] !((Rew.substs (#0 :> fun x => #x.succ.succ)).hom p.val)))”)
          ⋯).val
Theorem: Arith.Model.Definable.ball_lt
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
P : (Fin k → M) → M → Prop
f : (Fin k → M) → M
bf : HSemiformula L M (k + 1) (𝚺, m + 1)
hbf : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) bf
p : HSemiformula L M k.succ (𝚺, m + 1)
hp : DefinedWithParam (fun w => P (fun x => w x.succ) (w 0)) p
v : Fin k → M
⊢ (fun v => ∀ x < f v, P v x) v ↔
    (Semiformula.Evalm M v id)
      (HSemiformula.mkSigma (“∃' (!bf.val ∧ (∀[#0 < #1] !((Rew.substs (#0 :> fun x => #x.succ.succ)).hom p.val)))”)
          ⋯).val
Tactic:
simp [hbf.df.iff, hp.df.iff]
Proof state after:
no goals
Theorem: Arith.Model.Definable.ball_lt
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
P : (Fin k → M) → M → Prop
f : (Fin k → M) → M
bf : HSemiformula L M (k + 1) (𝚺, m + 1)
hbf : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) bf
p : HSemiformula L M k.succ (𝚷, m + 1)
hp : DefinedWithParam (fun w => P (fun x => w x.succ) (w 0)) p
⊢ Hierarchy 𝚷 (m + 1) (“∀' (!bf.val → (∀[#0 < #1] !((Rew.substs (#0 :> fun x => #x.succ.succ)).hom p.val)))”)
Tactic:
simp
Proof state after:
no goals
Theorem: Arith.Model.Definable.ball_lt
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
P : (Fin k → M) → M → Prop
f : (Fin k → M) → M
bf : HSemiformula L M (k + 1) (𝚺, m + 1)
hbf : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) bf
p : HSemiformula L M k.succ (𝚷, m + 1)
hp : DefinedWithParam (fun w => P (fun x => w x.succ) (w 0)) p
⊢ DefinedWithParam (fun v => ∀ x < f v, P v x)
    (HSemiformula.mkPi (“∀' (!bf.val → (∀[#0 < #1] !((Rew.substs (#0 :> fun x => #x.succ.succ)).hom 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
m : ℕ
Γ : SigmaPiDelta
P : (Fin k → M) → M → Prop
f : (Fin k → M) → M
bf : HSemiformula L M (k + 1) (𝚺, m + 1)
hbf : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) bf
p : HSemiformula L M k.succ (𝚷, m + 1)
hp : DefinedWithParam (fun w => P (fun x => w x.succ) (w 0)) p
v : Fin k → M
⊢ (fun v => ∀ x < f v, P v x) v ↔
    (Semiformula.Evalm M v id)
      (HSemiformula.mkPi (“∀' (!bf.val → (∀[#0 < #1] !((Rew.substs (#0 :> fun x => #x.succ.succ)).hom p.val)))”) ⋯).val
Theorem: Arith.Model.Definable.ball_lt
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
P : (Fin k → M) → M → Prop
f : (Fin k → M) → M
bf : HSemiformula L M (k + 1) (𝚺, m + 1)
hbf : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) bf
p : HSemiformula L M k.succ (𝚷, m + 1)
hp : DefinedWithParam (fun w => P (fun x => w x.succ) (w 0)) p
v : Fin k → M
⊢ (fun v => ∀ x < f v, P v x) v ↔
    (Semiformula.Evalm M v id)
      (HSemiformula.mkPi (“∀' (!bf.val → (∀[#0 < #1] !((Rew.substs (#0 :> fun x => #x.succ.succ)).hom p.val)))”) ⋯).val
Tactic:
simp [hbf.df.iff, hp.df.iff]
Proof state after:
no goals
Theorem: Arith.Model.Definable.ball_lt
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
P : (Fin k → M) → M → Prop
f : (Fin k → M) → M
bf : HSemiformula L M (k + 1) (𝚺, m + 1)
hbf : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) bf
p : HSemiformula L M k.succ (𝚫, m + 1)
hp : DefinedWithParam (fun w => P (fun x => w x.succ) (w 0)) p
⊢ Hierarchy 𝚺 (m + 1) (“∃' (!bf.val ∧ (∀[#0 < #1] !((Rew.substs (#0 :> fun x => #x.succ.succ)).hom p.sigma.val)))”)
Tactic:
simp
Proof state after:
no goals
Theorem: Arith.Model.Definable.ball_lt
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
P : (Fin k → M) → M → Prop
f : (Fin k → M) → M
bf : HSemiformula L M (k + 1) (𝚺, m + 1)
hbf : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) bf
p : HSemiformula L M k.succ (𝚫, m + 1)
hp : DefinedWithParam (fun w => P (fun x => w x.succ) (w 0)) p
⊢ DefinedWithParam (fun v => ∀ x < f v, P v x)
    (HSemiformula.mkSigma (“∃' (!bf.val ∧ (∀[#0 < #1] !((Rew.substs (#0 :> fun x => #x.succ.succ)).hom p.sigma.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
m : ℕ
Γ : SigmaPiDelta
P : (Fin k → M) → M → Prop
f : (Fin k → M) → M
bf : HSemiformula L M (k + 1) (𝚺, m + 1)
hbf : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) bf
p : HSemiformula L M k.succ (𝚫, m + 1)
hp : DefinedWithParam (fun w => P (fun x => w x.succ) (w 0)) p
v : Fin k → M
⊢ (fun v => ∀ x < f v, P v x) v ↔
    (Semiformula.Evalm M v id)
      (HSemiformula.mkSigma
          (“∃' (!bf.val ∧ (∀[#0 < #1] !((Rew.substs (#0 :> fun x => #x.succ.succ)).hom p.sigma.val)))”) ⋯).val
Theorem: Arith.Model.Definable.ball_lt
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
P : (Fin k → M) → M → Prop
f : (Fin k → M) → M
bf : HSemiformula L M (k + 1) (𝚺, m + 1)
hbf : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) bf
p : HSemiformula L M k.succ (𝚫, m + 1)
hp : DefinedWithParam (fun w => P (fun x => w x.succ) (w 0)) p
v : Fin k → M
⊢ (fun v => ∀ x < f v, P v x) v ↔
    (Semiformula.Evalm M v id)
      (HSemiformula.mkSigma
          (“∃' (!bf.val ∧ (∀[#0 < #1] !((Rew.substs (#0 :> fun x => #x.succ.succ)).hom p.sigma.val)))”) ⋯).val
Tactic:
simp [hbf.df.iff, hp.df.iff, HSemiformula.val_sigma]
Proof state after:
no goals
Theorem: Arith.Model.Definable.ball_lt
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
P : (Fin k → M) → M → Prop
f : (Fin k → M) → M
bf : HSemiformula L M (k + 1) (𝚺, m + 1)
hbf : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) bf
p : HSemiformula L M k.succ (𝚫, m + 1)
hp : DefinedWithParam (fun w => P (fun x => w x.succ) (w 0)) p
⊢ Hierarchy 𝚷 (m + 1) (“∀' (!bf.val → (∀[#0 < #1] !((Rew.substs (#0 :> fun x => #x.succ.succ)).hom p.pi.val)))”)
Tactic:
simp
Proof state after:
no goals
Theorem: Arith.Model.Definable.ball_lt
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
P : (Fin k → M) → M → Prop
f : (Fin k → M) → M
bf : HSemiformula L M (k + 1) (𝚺, m + 1)
hbf : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) bf
p : HSemiformula L M k.succ (𝚫, m + 1)
hp : DefinedWithParam (fun w => P (fun x => w x.succ) (w 0)) p
⊢ DefinedWithParam (fun v => ∀ x < f v, P v x)
    (HSemiformula.mkPi (“∀' (!bf.val → (∀[#0 < #1] !((Rew.substs (#0 :> fun x => #x.succ.succ)).hom p.pi.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
m : ℕ
Γ : SigmaPiDelta
P : (Fin k → M) → M → Prop
f : (Fin k → M) → M
bf : HSemiformula L M (k + 1) (𝚺, m + 1)
hbf : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) bf
p : HSemiformula L M k.succ (𝚫, m + 1)
hp : DefinedWithParam (fun w => P (fun x => w x.succ) (w 0)) p
v : Fin k → M
⊢ (fun v => ∀ x < f v, P v x) v ↔
    (Semiformula.Evalm M v id)
      (HSemiformula.mkPi (“∀' (!bf.val → (∀[#0 < #1] !((Rew.substs (#0 :> fun x => #x.succ.succ)).hom p.pi.val)))”)
          ⋯).val
Theorem: Arith.Model.Definable.ball_lt
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
P : (Fin k → M) → M → Prop
f : (Fin k → M) → M
bf : HSemiformula L M (k + 1) (𝚺, m + 1)
hbf : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) bf
p : HSemiformula L M k.succ (𝚫, m + 1)
hp : DefinedWithParam (fun w => P (fun x => w x.succ) (w 0)) p
v : Fin k → M
⊢ (fun v => ∀ x < f v, P v x) v ↔
    (Semiformula.Evalm M v id)
      (HSemiformula.mkPi (“∀' (!bf.val → (∀[#0 < #1] !((Rew.substs (#0 :> fun x => #x.succ.succ)).hom p.pi.val)))”)
          ⋯).val
Tactic:
simp [hbf.df.iff, hp.df.iff, hp.proper.iff']
Proof state after:
no goals
Theorem: Arith.Model.Definable.bex_lt
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
P : (Fin k → M) → M → Prop
f : (Fin k → M) → M
hf : DefinableFunction L (𝚺, m + 1) f
h : Definable L (Γ, m + 1) fun w => P (fun x => w x.succ) (w 0)
⊢ Definable L (Γ, m + 1) fun v => ∃ x < f v, P v x
Tactic:
rcases hf with ⟨bf, hbf⟩
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
P : (Fin k → M) → M → Prop
f : (Fin k → M) → M
h : Definable L (Γ, m + 1) fun w => P (fun x => w x.succ) (w 0)
bf : HSemiformula L M (k + 1) (𝚺, m + 1)
hbf : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) bf
⊢ Definable L (Γ, m + 1) fun v => ∃ x < f v, P v x
Theorem: Arith.Model.Definable.bex_lt
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
P : (Fin k → M) → M → Prop
f : (Fin k → M) → M
h : Definable L (Γ, m + 1) fun w => P (fun x => w x.succ) (w 0)
bf : HSemiformula L M (k + 1) (𝚺, m + 1)
hbf : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) bf
⊢ Definable L (Γ, m + 1) fun v => ∃ x < f v, P v x
Tactic:
rcases h with ⟨p, hp⟩
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 : (Fin k → M) → M → Prop
f : (Fin k → M) → M
bf : HSemiformula L M (k + 1) (𝚺, m + 1)
hbf : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) bf
p : HSemiformula L M k.succ (Γ, m + 1)
hp : DefinedWithParam (fun w => P (fun x => w x.succ) (w 0)) p
⊢ Definable L (Γ, m + 1) fun v => ∃ x < f v, P v x
Theorem: Arith.Model.Definable.bex_lt
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
P : (Fin k → M) → M → Prop
f : (Fin k → M) → M
bf : HSemiformula L M (k + 1) (𝚺, m + 1)
hbf : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) bf
p : HSemiformula L M k.succ (𝚺, m + 1)
hp : DefinedWithParam (fun w => P (fun x => w x.succ) (w 0)) p
⊢ Hierarchy 𝚺 (m + 1) (“∃' (!bf.val ∧ (∃[#0 < #1] !((Rew.substs (#0 :> fun x => #x.succ.succ)).hom p.val)))”)
Tactic:
simp
Proof state after:
no goals
Theorem: Arith.Model.Definable.bex_lt
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
P : (Fin k → M) → M → Prop
f : (Fin k → M) → M
bf : HSemiformula L M (k + 1) (𝚺, m + 1)
hbf : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) bf
p : HSemiformula L M k.succ (𝚺, m + 1)
hp : DefinedWithParam (fun w => P (fun x => w x.succ) (w 0)) p
⊢ DefinedWithParam (fun v => ∃ x < f v, P v x)
    (HSemiformula.mkSigma (“∃' (!bf.val ∧ (∃[#0 < #1] !((Rew.substs (#0 :> fun x => #x.succ.succ)).hom 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
m : ℕ
Γ : SigmaPiDelta
P : (Fin k → M) → M → Prop
f : (Fin k → M) → M
bf : HSemiformula L M (k + 1) (𝚺, m + 1)
hbf : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) bf
p : HSemiformula L M k.succ (𝚺, m + 1)
hp : DefinedWithParam (fun w => P (fun x => w x.succ) (w 0)) p
v : Fin k → M
⊢ (fun v => ∃ x < f v, P v x) v ↔
    (Semiformula.Evalm M v id)
      (HSemiformula.mkSigma (“∃' (!bf.val ∧ (∃[#0 < #1] !((Rew.substs (#0 :> fun x => #x.succ.succ)).hom p.val)))”)
          ⋯).val
Theorem: Arith.Model.Definable.bex_lt
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
P : (Fin k → M) → M → Prop
f : (Fin k → M) → M
bf : HSemiformula L M (k + 1) (𝚺, m + 1)
hbf : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) bf
p : HSemiformula L M k.succ (𝚺, m + 1)
hp : DefinedWithParam (fun w => P (fun x => w x.succ) (w 0)) p
v : Fin k → M
⊢ (fun v => ∃ x < f v, P v x) v ↔
    (Semiformula.Evalm M v id)
      (HSemiformula.mkSigma (“∃' (!bf.val ∧ (∃[#0 < #1] !((Rew.substs (#0 :> fun x => #x.succ.succ)).hom p.val)))”)
          ⋯).val
Tactic:
simp [hbf.df.iff, hp.df.iff]
Proof state after:
no goals
Theorem: Arith.Model.Definable.bex_lt
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
P : (Fin k → M) → M → Prop
f : (Fin k → M) → M
bf : HSemiformula L M (k + 1) (𝚺, m + 1)
hbf : DefinedWithParam (fun v => v 0 = f fun x => v x.succ) bf
p : HSemiformula L M k.succ (𝚷, m + 1)
hp : DefinedWithParam (fun w => P (fun x => w x.succ) (w 0)) p
⊢ Hierarchy 𝚷 (m + 1) (“∀' (!bf.val → (∃[#0 < #1] !((Rew.substs (#0 :> fun x => #x.succ.succ)).hom p.val)))”)
Tactic:
simp
Proof state after:
no goals