Theorem: Arith.Model.Definable.ball_le (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
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:
have : Definable L (Γ, m + 1) (fun v ↦ ∀ x < f v + 1, P v x) := ball_lt (DefinableFunction₂.comp (by simp) hf (by simp)) h
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
hf : DefinableFunction L (𝚺, m + 1) f
h : Definable L (Γ, m + 1) fun w => P (fun x => w x.succ) (w 0)
this : Definable L (Γ, m + 1) fun v => ∀ x < f v + 1, P v x
⊢ Definable L (Γ, m + 1) fun v => ∀ x ≤ f v, P v x

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)
this : Definable L (Γ, m + 1) fun v => ∀ x < f v + 1, P v x
⊢ Definable L (Γ, m + 1) fun v => ∀ x ≤ f v, P v x
Tactic:
exact this.of_iff <| by intro v; simp [lt_succ_iff_le]
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
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)
⊢ DefinableFunction₂ L (𝚺, m + 1) HAdd.hAdd
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
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)
⊢ DefinableFunction L (𝚺, m + 1) fun v => 1
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
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)
this : Definable L (Γ, m + 1) fun v => ∀ x < f v + 1, P v x
⊢ ∀ (x : Fin k → M), (∀ x_1 ≤ f x, P x x_1) ↔ ∀ x_1 < f x + 1, P x x_1
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
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)
this : Definable L (Γ, m + 1) fun v => ∀ x < f v + 1, P v x
v : Fin k → M
⊢ (∀ x ≤ f v, P v x) ↔ ∀ x < f v + 1, P v x

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)
this : Definable L (Γ, m + 1) fun v => ∀ x < f v + 1, P v x
v : Fin k → M
⊢ (∀ x ≤ f v, P v x) ↔ ∀ x < f v + 1, P v x
Tactic:
simp [lt_succ_iff_le]
State after:
no goals