Boldface

def LO.FirstOrder.Defined {V : Type u_1} {L : Language} {k : ℕ} (R : (Fin k → V) → Prop) [Structure L V] (φ : Semisentence L k) : Prop

Imported declaration from the Incompleteness formalization.

Equations

• LO.FirstOrder.Defined R φ = ∀ (v : Fin k → V), R v ↔ V ⊧/v φ

def LO.FirstOrder.DefinedWithParam {V : Type u_1} {L : Language} {k : ℕ} (R : (Fin k → V) → Prop) [Structure L V] (φ : Semiformula L V k) : Prop

Imported declaration from the Incompleteness formalization.

Equations

• LO.FirstOrder.DefinedWithParam R φ = ∀ (v : Fin k → V), R v ↔ (LO.FirstOrder.Semiformula.Evalm V v id) φ

theorem LO.FirstOrder.Defined.iff {L : Language} {V : Type u_2} [Structure L V] {k : ℕ} {R : (Fin k → V) → Prop} {φ : Semisentence L k} (h : Defined R φ) (v : Fin k → V) : V ⊧/v φ ↔ R v

theorem LO.FirstOrder.DefinedWithParam.iff {L : Language} {V : Type u_2} [Structure L V] {k : ℕ} {R : (Fin k → V) → Prop} {φ : Semiformula L V k} (h : DefinedWithParam R φ) (v : Fin k → V) : (Semiformula.Evalm V v id) φ ↔ R v

def LO.FirstOrder.Arith.HierarchySymbol.Defined {V : Type u_2} [ORingStruc V] {k : ℕ} (R : (Fin k → V) → Prop) {ℌ : HierarchySymbol} : ℌ.Semisentence k → Prop

Imported declaration from the Incompleteness formalization.

Equations

• One or more equations did not get rendered due to their size.
• LO.FirstOrder.Arith.HierarchySymbol.Defined R φ = LO.FirstOrder.Defined R (LO.FirstOrder.Arith.HierarchySymbol.Semiformula.val φ)
• LO.FirstOrder.Arith.HierarchySymbol.Defined R φ = LO.FirstOrder.Defined R (LO.FirstOrder.Arith.HierarchySymbol.Semiformula.val φ)

def LO.FirstOrder.Arith.HierarchySymbol.DefinedWithParam {V : Type u_2} [ORingStruc V] {k : ℕ} (R : (Fin k → V) → Prop) {ℌ : HierarchySymbol} : HierarchySymbol.Semiformula V k ℌ → Prop

Imported declaration from the Incompleteness formalization.

Equations

• LO.FirstOrder.Arith.HierarchySymbol.DefinedWithParam R φ = LO.FirstOrder.DefinedWithParam R φ.val
• LO.FirstOrder.Arith.HierarchySymbol.DefinedWithParam R φ = LO.FirstOrder.DefinedWithParam R φ.val
• LO.FirstOrder.Arith.HierarchySymbol.DefinedWithParam R φ = (LO.FirstOrder.Arith.HierarchySymbol.Semiformula.ProperWithParamOn V φ ∧ LO.FirstOrder.DefinedWithParam R φ.val)

class LO.FirstOrder.Arith.HierarchySymbol.Lightface {V : Type u_2} [ORingStruc V] (ℌ : HierarchySymbol) {k : ℕ} (P : (Fin k → V) → Prop) : Prop

Imported declaration from the Incompleteness formalization.

• definable : ∃ (φ : ℌ.Semisentence k), Defined P φ

class LO.FirstOrder.Arith.HierarchySymbol.Boldface {V : Type u_2} [ORingStruc V] (ℌ : HierarchySymbol) {k : ℕ} (P : (Fin k → V) → Prop) : Prop

Imported declaration from the Incompleteness formalization.

• definable : ∃ (φ : HierarchySymbol.Semiformula V k ℌ), DefinedWithParam P φ

@[reducible, inline]

abbrev LO.FirstOrder.Arith.HierarchySymbol.DefinedPred {V : Type u_2} [ORingStruc V] (ℌ : HierarchySymbol) (P : V → Prop) (φ : ℌ.Semisentence 1) : Prop

Imported declaration from the Incompleteness formalization.

Equations

• (ℌ-Predicate P via φ) = LO.FirstOrder.Arith.HierarchySymbol.Defined (fun (v : Fin 1 → V) => P (v 0)) φ

@[reducible, inline]

abbrev LO.FirstOrder.Arith.HierarchySymbol.DefinedRel {V : Type u_2} [ORingStruc V] (ℌ : HierarchySymbol) (R : V → V → Prop) (φ : ℌ.Semisentence 2) : Prop

Imported declaration from the Incompleteness formalization.

Equations

• (ℌ-Relation R via φ) = LO.FirstOrder.Arith.HierarchySymbol.Defined (fun (v : Fin 2 → V) => R (v 0) (v 1)) φ

@[reducible, inline]

abbrev LO.FirstOrder.Arith.HierarchySymbol.DefinedRel₃ {V : Type u_2} [ORingStruc V] (ℌ : HierarchySymbol) (R : V → V → V → Prop) (φ : ℌ.Semisentence 3) : Prop

Imported declaration from the Incompleteness formalization.

Equations

• (ℌ-Relation₃ R via φ) = LO.FirstOrder.Arith.HierarchySymbol.Defined (fun (v : Fin 3 → V) => R (v 0) (v 1) (v 2)) φ

@[reducible, inline]

abbrev LO.FirstOrder.Arith.HierarchySymbol.DefinedRel₄ {V : Type u_2} [ORingStruc V] (ℌ : HierarchySymbol) (R : V → V → V → V → Prop) (φ : ℌ.Semisentence 4) : Prop

Imported declaration from the Incompleteness formalization.

Equations

• (ℌ-Relation₄ R via φ) = LO.FirstOrder.Arith.HierarchySymbol.Defined (fun (v : Fin 4 → V) => R (v 0) (v 1) (v 2) (v 3)) φ

@[reducible, inline]

abbrev LO.FirstOrder.Arith.HierarchySymbol.DefinedFunction {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} {k : ℕ} (f : (Fin k → V) → V) (φ : ℌ.Semisentence (k + 1)) : Prop

Imported declaration from the Incompleteness formalization.

Equations

• LO.FirstOrder.Arith.HierarchySymbol.DefinedFunction f φ = LO.FirstOrder.Arith.HierarchySymbol.Defined (fun (v : Fin (k + 1) → V) => v 0 = f fun (x : Fin k) => v x.succ) φ

@[reducible, inline]

abbrev LO.FirstOrder.Arith.HierarchySymbol.DefinedFunction₀ {V : Type u_2} [ORingStruc V] (ℌ : HierarchySymbol) (c : V) (φ : ℌ.Semisentence 1) : Prop

Imported declaration from the Incompleteness formalization.

Equations

• (ℌ-Function₀ c via φ) = LO.FirstOrder.Arith.HierarchySymbol.DefinedFunction (fun (x : Fin 0 → V) => c) φ

@[reducible, inline]

abbrev LO.FirstOrder.Arith.HierarchySymbol.DefinedFunction₁ {V : Type u_2} [ORingStruc V] (ℌ : HierarchySymbol) (f : V → V) (φ : ℌ.Semisentence 2) : Prop

Imported declaration from the Incompleteness formalization.

Equations

• (ℌ-Function₁ f via φ) = LO.FirstOrder.Arith.HierarchySymbol.DefinedFunction (fun (v : Fin 1 → V) => f (v 0)) φ

@[reducible, inline]

abbrev LO.FirstOrder.Arith.HierarchySymbol.DefinedFunction₂ {V : Type u_2} [ORingStruc V] (ℌ : HierarchySymbol) (f : V → V → V) (φ : ℌ.Semisentence 3) : Prop

Imported declaration from the Incompleteness formalization.

Equations

• (ℌ-Function₂ f via φ) = LO.FirstOrder.Arith.HierarchySymbol.DefinedFunction (fun (v : Fin 2 → V) => f (v 0) (v 1)) φ

@[reducible, inline]

abbrev LO.FirstOrder.Arith.HierarchySymbol.DefinedFunction₃ {V : Type u_2} [ORingStruc V] (ℌ : HierarchySymbol) (f : V → V → V → V) (φ : ℌ.Semisentence 4) : Prop

Imported declaration from the Incompleteness formalization.

Equations

• (ℌ-Function₃ f via φ) = LO.FirstOrder.Arith.HierarchySymbol.DefinedFunction (fun (v : Fin 3 → V) => f (v 0) (v 1) (v 2)) φ

@[reducible, inline]

abbrev LO.FirstOrder.Arith.HierarchySymbol.DefinedFunction₄ {V : Type u_2} [ORingStruc V] (ℌ : HierarchySymbol) (f : V → V → V → V → V) (φ : ℌ.Semisentence 5) : Prop

Imported declaration from the Incompleteness formalization.

Equations

• (ℌ-Function₄ f via φ) = LO.FirstOrder.Arith.HierarchySymbol.DefinedFunction (fun (v : Fin 4 → V) => f (v 0) (v 1) (v 2) (v 3)) φ

@[reducible, inline]

abbrev LO.FirstOrder.Arith.HierarchySymbol.DefinedFunction₅ {V : Type u_2} [ORingStruc V] (ℌ : HierarchySymbol) (f : V → V → V → V → V → V) (φ : ℌ.Semisentence 6) : Prop

Imported declaration from the Incompleteness formalization.

Equations

• (ℌ-Function₅ f via φ) = LO.FirstOrder.Arith.HierarchySymbol.DefinedFunction (fun (v : Fin 5 → V) => f (v 0) (v 1) (v 2) (v 3) (v 4)) φ

@[reducible, inline]

abbrev LO.FirstOrder.Arith.HierarchySymbol.BoldfacePred {V : Type u_2} [ORingStruc V] (ℌ : HierarchySymbol) (P : V → Prop) : Prop

Imported declaration from the Incompleteness formalization.

Equations

• (ℌ-Predicate P) = ℌ.Boldface fun (v : Fin 1 → V) => P (v 0)

@[reducible, inline]

abbrev LO.FirstOrder.Arith.HierarchySymbol.BoldfaceRel {V : Type u_2} [ORingStruc V] (ℌ : HierarchySymbol) (P : V → V → Prop) : Prop

Imported declaration from the Incompleteness formalization.

Equations

• (ℌ-Relation P) = ℌ.Boldface fun (v : Fin 2 → V) => P (v 0) (v 1)

@[reducible, inline]

abbrev LO.FirstOrder.Arith.HierarchySymbol.BoldfaceRel₃ {V : Type u_2} [ORingStruc V] (ℌ : HierarchySymbol) (P : V → V → V → Prop) : Prop

Imported declaration from the Incompleteness formalization.

Equations

• (ℌ-Relation₃ P) = ℌ.Boldface fun (v : Fin 3 → V) => P (v 0) (v 1) (v 2)

@[reducible, inline]

abbrev LO.FirstOrder.Arith.HierarchySymbol.BoldfaceRel₄ {V : Type u_2} [ORingStruc V] (ℌ : HierarchySymbol) (P : V → V → V → V → Prop) : Prop

Imported declaration from the Incompleteness formalization.

Equations

• (ℌ-Relation₄ P) = ℌ.Boldface fun (v : Fin 4 → V) => P (v 0) (v 1) (v 2) (v 3)

@[reducible, inline]

abbrev LO.FirstOrder.Arith.HierarchySymbol.BoldfaceRel₅ {V : Type u_2} [ORingStruc V] (ℌ : HierarchySymbol) (P : V → V → V → V → V → Prop) : Prop

Imported declaration from the Incompleteness formalization.

Equations

• (ℌ-Relation₅ P) = ℌ.Boldface fun (v : Fin 5 → V) => P (v 0) (v 1) (v 2) (v 3) (v 4)

@[reducible, inline]

abbrev LO.FirstOrder.Arith.HierarchySymbol.BoldfaceRel₆ {V : Type u_2} [ORingStruc V] (ℌ : HierarchySymbol) (P : V → V → V → V → V → V → Prop) : Prop

Imported declaration from the Incompleteness formalization.

Equations

• ℌ.BoldfaceRel₆ P = ℌ.Boldface fun (v : Fin 6 → V) => P (v 0) (v 1) (v 2) (v 3) (v 4) (v 5)

@[reducible, inline]

abbrev LO.FirstOrder.Arith.HierarchySymbol.BoldfaceFunction {V : Type u_2} [ORingStruc V] (ℌ : HierarchySymbol) {k : ℕ} (f : (Fin k → V) → V) : Prop

Imported declaration from the Incompleteness formalization.

Equations

• ℌ.BoldfaceFunction f = ℌ.Boldface fun (v : Fin (k + 1) → V) => v 0 = f fun (x : Fin k) => v x.succ

@[reducible, inline]

abbrev LO.FirstOrder.Arith.HierarchySymbol.BoldfaceFunction₀ {V : Type u_2} [ORingStruc V] (ℌ : HierarchySymbol) (c : V) : Prop

Imported declaration from the Incompleteness formalization.

Equations

• ℌ.BoldfaceFunction₀ c = ℌ.BoldfaceFunction fun (x : Fin 0 → V) => c

@[reducible, inline]

abbrev LO.FirstOrder.Arith.HierarchySymbol.BoldfaceFunction₁ {V : Type u_2} [ORingStruc V] (ℌ : HierarchySymbol) (f : V → V) : Prop

Imported declaration from the Incompleteness formalization.

Equations

• (ℌ-Function₁ f) = ℌ.BoldfaceFunction fun (v : Fin 1 → V) => f (v 0)

@[reducible, inline]

abbrev LO.FirstOrder.Arith.HierarchySymbol.BoldfaceFunction₂ {V : Type u_2} [ORingStruc V] (ℌ : HierarchySymbol) (f : V → V → V) : Prop

Imported declaration from the Incompleteness formalization.

Equations

• (ℌ-Function₂ f) = ℌ.BoldfaceFunction fun (v : Fin 2 → V) => f (v 0) (v 1)

@[reducible, inline]

abbrev LO.FirstOrder.Arith.HierarchySymbol.BoldfaceFunction₃ {V : Type u_2} [ORingStruc V] (ℌ : HierarchySymbol) (f : V → V → V → V) : Prop

Imported declaration from the Incompleteness formalization.

Equations

• (ℌ-Function₃ f) = ℌ.BoldfaceFunction fun (v : Fin 3 → V) => f (v 0) (v 1) (v 2)

@[reducible, inline]

abbrev LO.FirstOrder.Arith.HierarchySymbol.BoldfaceFunction₄ {V : Type u_2} [ORingStruc V] (ℌ : HierarchySymbol) (f : V → V → V → V → V) : Prop

Imported declaration from the Incompleteness formalization.

Equations

• (ℌ-Function₄ f) = ℌ.BoldfaceFunction fun (v : Fin 4 → V) => f (v 0) (v 1) (v 2) (v 3)

@[reducible, inline]

abbrev LO.FirstOrder.Arith.HierarchySymbol.BoldfaceFunction₅ {V : Type u_2} [ORingStruc V] (ℌ : HierarchySymbol) (f : V → V → V → V → V → V) : Prop

Imported declaration from the Incompleteness formalization.

Equations

• ℌ.BoldfaceFunction₅ f = ℌ.BoldfaceFunction fun (v : Fin 5 → V) => f (v 0) (v 1) (v 2) (v 3) (v 4)

def LO.FirstOrder.Arith.HierarchySymbol.«term_-Predicate_Via_» : Lean.TrailingParserDescr

Imported declaration from the Incompleteness formalization.

Equations

• One or more equations did not get rendered due to their size.

def LO.FirstOrder.Arith.HierarchySymbol.«term_-Relation_Via_» : Lean.TrailingParserDescr

Imported declaration from the Incompleteness formalization.

Equations

• One or more equations did not get rendered due to their size.

def LO.FirstOrder.Arith.HierarchySymbol.«term_-Relation₃_Via_» : Lean.TrailingParserDescr

Imported declaration from the Incompleteness formalization.

Equations

• One or more equations did not get rendered due to their size.

def LO.FirstOrder.Arith.HierarchySymbol.«term_-Relation₄_Via_» : Lean.TrailingParserDescr

Imported declaration from the Incompleteness formalization.

Equations

• One or more equations did not get rendered due to their size.

def LO.FirstOrder.Arith.HierarchySymbol.«term_-Function₀_Via_» : Lean.TrailingParserDescr

Imported declaration from the Incompleteness formalization.

Equations

• One or more equations did not get rendered due to their size.

def LO.FirstOrder.Arith.HierarchySymbol.«term_-Function₁_Via_» : Lean.TrailingParserDescr

Imported declaration from the Incompleteness formalization.

Equations

• One or more equations did not get rendered due to their size.

def LO.FirstOrder.Arith.HierarchySymbol.«term_-Function₂_Via_» : Lean.TrailingParserDescr

Imported declaration from the Incompleteness formalization.

Equations

• One or more equations did not get rendered due to their size.

def LO.FirstOrder.Arith.HierarchySymbol.«term_-Function₃_Via_» : Lean.TrailingParserDescr

Imported declaration from the Incompleteness formalization.

Equations

• One or more equations did not get rendered due to their size.

def LO.FirstOrder.Arith.HierarchySymbol.«term_-Function₄_Via_» : Lean.TrailingParserDescr

Imported declaration from the Incompleteness formalization.

Equations

• One or more equations did not get rendered due to their size.

def LO.FirstOrder.Arith.HierarchySymbol.«term_-Function₅_Via_» : Lean.TrailingParserDescr

Imported declaration from the Incompleteness formalization.

Equations

• One or more equations did not get rendered due to their size.

def LO.FirstOrder.Arith.HierarchySymbol.«term_-Predicate_» : Lean.TrailingParserDescr

Imported declaration from the Incompleteness formalization.

Equations

• One or more equations did not get rendered due to their size.

def LO.FirstOrder.Arith.HierarchySymbol.«term_-Relation_» : Lean.TrailingParserDescr

Imported declaration from the Incompleteness formalization.

Equations

• One or more equations did not get rendered due to their size.

def LO.FirstOrder.Arith.HierarchySymbol.«term_-Relation₃_» : Lean.TrailingParserDescr

Imported declaration from the Incompleteness formalization.

Equations

• One or more equations did not get rendered due to their size.

def LO.FirstOrder.Arith.HierarchySymbol.«term_-Relation₄_» : Lean.TrailingParserDescr

Imported declaration from the Incompleteness formalization.

Equations

• One or more equations did not get rendered due to their size.

def LO.FirstOrder.Arith.HierarchySymbol.«term_-Relation₅_» : Lean.TrailingParserDescr

Imported declaration from the Incompleteness formalization.

Equations

• One or more equations did not get rendered due to their size.

def LO.FirstOrder.Arith.HierarchySymbol.«term_-Function₁_» : Lean.TrailingParserDescr

Imported declaration from the Incompleteness formalization.

Equations

• One or more equations did not get rendered due to their size.

def LO.FirstOrder.Arith.HierarchySymbol.«term_-Function₂_» : Lean.TrailingParserDescr

Imported declaration from the Incompleteness formalization.

Equations

• One or more equations did not get rendered due to their size.

def LO.FirstOrder.Arith.HierarchySymbol.«term_-Function₃_» : Lean.TrailingParserDescr

Imported declaration from the Incompleteness formalization.

Equations

• One or more equations did not get rendered due to their size.

def LO.FirstOrder.Arith.HierarchySymbol.«term_-Function₄_» : Lean.TrailingParserDescr

Imported declaration from the Incompleteness formalization.

Equations

• One or more equations did not get rendered due to their size.

theorem LO.FirstOrder.Arith.HierarchySymbol.Defined.df {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} {k : ℕ} {R : (Fin k → V) → Prop} {φ : ℌ.Semisentence k} (h : Defined R φ) : FirstOrder.Defined R (Semiformula.val φ)

theorem LO.FirstOrder.Arith.HierarchySymbol.Defined.proper {V : Type u_2} [ORingStruc V] {k : ℕ} {R : (Fin k → V) → Prop} {m : ℕ} {φ : { Γ := Dlt, rank := m }.Semisentence k} (h : Defined R φ) : Semiformula.ProperOn V φ

theorem LO.FirstOrder.Arith.HierarchySymbol.Defined.of_zero {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} {k : ℕ} {R : (Fin k → V) → Prop} {φ : Sg0.Semisentence k} (h : Defined R φ) : Defined R (Semiformula.ofZero φ ℌ)

theorem LO.FirstOrder.Arith.HierarchySymbol.Defined.emb {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} {k : ℕ} {R : (Fin k → V) → Prop} {φ : ℌ.Semisentence k} (h : Defined R φ) : Defined R (Semiformula.emb φ)

theorem LO.FirstOrder.Arith.HierarchySymbol.Defined.of_iff {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} {k : ℕ} {P Q : (Fin k → V) → Prop} (h : ∀ (x : Fin k → V), P x ↔ Q x) {φ : ℌ.Semisentence k} (H : Defined Q φ) : Defined P φ

theorem LO.FirstOrder.Arith.HierarchySymbol.Defined.to_definable {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} {k : ℕ} {P : (Fin k → V) → Prop} (φ : ℌ.Semisentence k) (hP : Defined P φ) : ℌ.Boldface P

theorem LO.FirstOrder.Arith.HierarchySymbol.Defined.to_definable₀ {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} {k : ℕ} {P : (Fin k → V) → Prop} {φ : Sg0.Semisentence k} (hP : Defined P φ) : ℌ.Boldface P

theorem LO.FirstOrder.Arith.HierarchySymbol.Defined.to_definable_oRing {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} {k : ℕ} {P : (Fin k → V) → Prop} (φ : ℌ.Semisentence k) (hP : Defined P φ) : ℌ.Boldface P

theorem LO.FirstOrder.Arith.HierarchySymbol.Defined.to_definable_oRing₀ {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} {k : ℕ} {P : (Fin k → V) → Prop} (φ : Sg0.Semisentence k) (hP : Defined P φ) : ℌ.Boldface P

theorem LO.FirstOrder.Arith.HierarchySymbol.DefinedFunction.of_eq {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} {k : ℕ} {f g : (Fin k → V) → V} (h : ∀ (x : Fin k → V), f x = g x) {φ : ℌ.Semisentence (k + 1)} (H : DefinedFunction f φ) : DefinedFunction g φ

theorem LO.FirstOrder.Arith.HierarchySymbol.DefinedFunction.graph_delta {V : Type u_2} [ORingStruc V] {k m : ℕ} {f : (Fin k → V) → V} {φ : { Γ := Sg, rank := m }.Semisentence (k + 1)} (h : DefinedFunction f φ) : DefinedFunction f (Semiformula.graphDelta φ)

theorem LO.FirstOrder.Arith.HierarchySymbol.DefinedWithParam.df {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} {k : ℕ} {R : (Fin k → V) → Prop} {φ : HierarchySymbol.Semiformula V k ℌ} (h : DefinedWithParam R φ) : FirstOrder.DefinedWithParam R φ.val

theorem LO.FirstOrder.Arith.HierarchySymbol.DefinedWithParam.proper {V : Type u_2} [ORingStruc V] {k : ℕ} {R : (Fin k → V) → Prop} {m : ℕ} {φ : HierarchySymbol.Semiformula V k { Γ := Dlt, rank := m }} (h : DefinedWithParam R φ) : Semiformula.ProperWithParamOn V φ

theorem LO.FirstOrder.Arith.HierarchySymbol.DefinedWithParam.of_zero {V : Type u_2} [ORingStruc V] {k : ℕ} {R : (Fin k → V) → Prop} {Γ' : SigmaPiDelta} {φ : HierarchySymbol.Semiformula V k { Γ := Γ', rank := 0 }} (h : DefinedWithParam R φ) {Γ : HierarchySymbol} : DefinedWithParam R (φ.ofZero Γ)

theorem LO.FirstOrder.Arith.HierarchySymbol.DefinedWithParam.of_deltaOne {V : Type u_2} [ORingStruc V] {k : ℕ} {R : (Fin k → V) → Prop} {Γ : SigmaPiDelta} {m : ℕ} {φ : HierarchySymbol.Semiformula V k Dlt1} (h : DefinedWithParam R φ) : DefinedWithParam R (φ.ofDeltaOne Γ m)

theorem LO.FirstOrder.Arith.HierarchySymbol.DefinedWithParam.emb {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} {k : ℕ} {R : (Fin k → V) → Prop} {φ : HierarchySymbol.Semiformula V k ℌ} (h : DefinedWithParam R φ) : DefinedWithParam R φ.emb

theorem LO.FirstOrder.Arith.HierarchySymbol.DefinedWithParam.of_iff {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} {k : ℕ} {P Q : (Fin k → V) → Prop} (h : ∀ (x : Fin k → V), P x ↔ Q x) {φ : HierarchySymbol.Semiformula V k ℌ} (H : DefinedWithParam Q φ) : DefinedWithParam P φ

theorem LO.FirstOrder.Arith.HierarchySymbol.DefinedWithParam.to_definable {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} {k : ℕ} {P : (Fin k → V) → Prop} {φ : HierarchySymbol.Semiformula V k ℌ} (h : DefinedWithParam P φ) : ℌ.Boldface P

theorem LO.FirstOrder.Arith.HierarchySymbol.DefinedWithParam.to_definable₀ {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} {k : ℕ} {P : (Fin k → V) → Prop} {Γ' : SigmaPiDelta} {φ : HierarchySymbol.Semiformula V k { Γ := Γ', rank := 0 }} (h : DefinedWithParam P φ) : ℌ.Boldface P

theorem LO.FirstOrder.Arith.HierarchySymbol.DefinedWithParam.to_definable_deltaOne {V : Type u_2} [ORingStruc V] {k : ℕ} {P : (Fin k → V) → Prop} {φ : HierarchySymbol.Semiformula V k Dlt1} {Γ : SigmaPiDelta} {m : ℕ} (h : DefinedWithParam P φ) : { Γ := Γ, rank := m + 1 }.Boldface P

theorem LO.FirstOrder.Arith.HierarchySymbol.DefinedWithParam.retraction {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} {k : ℕ} {P : (Fin k → V) → Prop} {l : ℕ} {φ : HierarchySymbol.Semiformula V k ℌ} (hp : DefinedWithParam P φ) (f : Fin k → Fin l) : DefinedWithParam (fun (v : Fin l → V) => P fun (i : Fin k) => v (f i)) (Semiformula.rew (Rew.substs fun (x : Fin k) => Semiterm.bvar (f x)) φ)

@[simp]

theorem LO.FirstOrder.Arith.HierarchySymbol.DefinedWithParam.verum {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} {k : ℕ} : DefinedWithParam (fun (x : Fin k → V) => True) ⊤

@[simp]

theorem LO.FirstOrder.Arith.HierarchySymbol.DefinedWithParam.falsum {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} {k : ℕ} : DefinedWithParam (fun (x : Fin k → V) => False) ⊥

theorem LO.FirstOrder.Arith.HierarchySymbol.DefinedWithParam.and {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} {k : ℕ} {P Q : (Fin k → V) → Prop} {φ ψ : HierarchySymbol.Semiformula V k ℌ} (hp : DefinedWithParam P φ) (hq : DefinedWithParam Q ψ) : DefinedWithParam (fun (x : Fin k → V) => P x ∧ Q x) (φ ⋏ ψ)

theorem LO.FirstOrder.Arith.HierarchySymbol.DefinedWithParam.or {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} {k : ℕ} {P Q : (Fin k → V) → Prop} {φ ψ : HierarchySymbol.Semiformula V k ℌ} (hp : DefinedWithParam P φ) (hq : DefinedWithParam Q ψ) : DefinedWithParam (fun (x : Fin k → V) => P x ∨ Q x) (φ ⋎ ψ)

theorem LO.FirstOrder.Arith.HierarchySymbol.DefinedWithParam.negSigma {V : Type u_2} [ORingStruc V] {k : ℕ} {P : (Fin k → V) → Prop} {m : ℕ} {φ : HierarchySymbol.Semiformula V k { Γ := Sg, rank := m }} (hp : DefinedWithParam P φ) : DefinedWithParam (fun (x : Fin k → V) => ¬P x) φ.negSigma

theorem LO.FirstOrder.Arith.HierarchySymbol.DefinedWithParam.negPi {V : Type u_2} [ORingStruc V] {k : ℕ} {P : (Fin k → V) → Prop} {m : ℕ} {φ : HierarchySymbol.Semiformula V k { Γ := Pg, rank := m }} (hp : DefinedWithParam P φ) : DefinedWithParam (fun (x : Fin k → V) => ¬P x) φ.negPi

theorem LO.FirstOrder.Arith.HierarchySymbol.DefinedWithParam.not {V : Type u_2} [ORingStruc V] {k : ℕ} {P : (Fin k → V) → Prop} {m : ℕ} {φ : HierarchySymbol.Semiformula V k { Γ := Dlt, rank := m }} (hp : DefinedWithParam P φ) : DefinedWithParam (fun (x : Fin k → V) => ¬P x) (∼φ)

theorem LO.FirstOrder.Arith.HierarchySymbol.DefinedWithParam.imp {V : Type u_2} [ORingStruc V] {k : ℕ} {P Q : (Fin k → V) → Prop} {m : ℕ} {φ ψ : HierarchySymbol.Semiformula V k { Γ := Dlt, rank := m }} (hp : DefinedWithParam P φ) (hq : DefinedWithParam Q ψ) : DefinedWithParam (fun (x : Fin k → V) => P x → Q x) (φ ==> ψ)

theorem LO.FirstOrder.Arith.HierarchySymbol.DefinedWithParam.iff {V : Type u_2} [ORingStruc V] {k : ℕ} {P Q : (Fin k → V) → Prop} {m : ℕ} {φ ψ : HierarchySymbol.Semiformula V k { Γ := Dlt, rank := m }} (hp : DefinedWithParam P φ) (hq : DefinedWithParam Q ψ) : DefinedWithParam (fun (x : Fin k → V) => P x ↔ Q x) (φ <=> ψ)

theorem LO.FirstOrder.Arith.HierarchySymbol.DefinedWithParam.ball {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} {k : ℕ} {P : (Fin (k + 1) → V) → Prop} {φ : HierarchySymbol.Semiformula V (k + 1) ℌ} (hp : DefinedWithParam P φ) (t : Semiterm ℒₒᵣ V k) : DefinedWithParam (fun (v : Fin k → V) => ∀ x < Semiterm.valm V v id t, P (x :> v)) (Semiformula.ball t φ)

theorem LO.FirstOrder.Arith.HierarchySymbol.DefinedWithParam.bex {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} {k : ℕ} {P : (Fin (k + 1) → V) → Prop} {φ : HierarchySymbol.Semiformula V (k + 1) ℌ} (hp : DefinedWithParam P φ) (t : Semiterm ℒₒᵣ V k) : DefinedWithParam (fun (v : Fin k → V) => ∃ x < Semiterm.valm V v id t, P (x :> v)) (Semiformula.bex t φ)

theorem LO.FirstOrder.Arith.HierarchySymbol.DefinedWithParam.ex {V : Type u_2} [ORingStruc V] {k m : ℕ} {P : (Fin (k + 1) → V) → Prop} {φ : HierarchySymbol.Semiformula V (k + 1) { Γ := Sg, rank := m + 1 }} (hp : DefinedWithParam P φ) : DefinedWithParam (fun (v : Fin k → V) => ∃ (x : V), P (x :> v)) φ.ex

theorem LO.FirstOrder.Arith.HierarchySymbol.DefinedWithParam.all {V : Type u_2} [ORingStruc V] {k m : ℕ} {P : (Fin (k + 1) → V) → Prop} {φ : HierarchySymbol.Semiformula V (k + 1) { Γ := Pg, rank := m + 1 }} (hp : DefinedWithParam P φ) : DefinedWithParam (fun (v : Fin k → V) => ∀ (x : V), P (x :> v)) φ.all

@[simp]

instance LO.FirstOrder.Arith.HierarchySymbol.BoldfaceRel.eq {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} : ℌ-Relation Eq

@[simp]

instance LO.FirstOrder.Arith.HierarchySymbol.BoldfaceRel.lt {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} : ℌ-Relation LT.lt

@[simp]

instance LO.FirstOrder.Arith.HierarchySymbol.BoldfaceRel.le {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} [V ⊧ₘ* 𝐏𝐀⁻] : ℌ-Relation LE.le

instance LO.FirstOrder.Arith.HierarchySymbol.BoldfaceFunction₂.add {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} : ℌ-Function₂ fun (x1 x2 : V) => x1 + x2

instance LO.FirstOrder.Arith.HierarchySymbol.BoldfaceFunction₂.mul {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} : ℌ-Function₂ fun (x1 x2 : V) => x1 * x2

instance LO.FirstOrder.Arith.HierarchySymbol.BoldfaceFunction₂.hAdd {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} : ℌ-Function₂ HAdd.hAdd

instance LO.FirstOrder.Arith.HierarchySymbol.BoldfaceFunction₂.hMul {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} : ℌ-Function₂ HMul.hMul

theorem LO.FirstOrder.Arith.HierarchySymbol.Boldface.mkPolarity {V : Type u_2} [ORingStruc V] {k m : ℕ} {P : (Fin k → V) → Prop} {Γ : Polarity} (φ : Semiformula ℒₒᵣ V k) (hp : Hierarchy Γ m φ) (hP : ∀ (v : Fin k → V), P v ↔ (Semiformula.Evalm V v id) φ) : { Γ := Γ.coe, rank := m }.Boldface P

theorem LO.FirstOrder.Arith.HierarchySymbol.Boldface.of_iff {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} {k : ℕ} {P Q : (Fin k → V) → Prop} (H : ℌ.Boldface Q) (h : ∀ (x : Fin k → V), P x ↔ Q x) : ℌ.Boldface P

theorem LO.FirstOrder.Arith.HierarchySymbol.Boldface.of_oRing {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} {k : ℕ} {P : (Fin k → V) → Prop} (h : ℌ.Boldface P) : ℌ.Boldface P

theorem LO.FirstOrder.Arith.HierarchySymbol.Boldface.of_delta {V : Type u_2} [ORingStruc V] {Γ : SigmaPiDelta} {k : ℕ} {P : (Fin k → V) → Prop} {m : ℕ} (h : { Γ := Dlt, rank := m }.Boldface P) : { Γ := Γ, rank := m }.Boldface P

instance LO.FirstOrder.Arith.HierarchySymbol.Boldface.instMkOfDeltaSigmaPiDelta {V : Type u_2} [ORingStruc V] {k : ℕ} {P : (Fin k → V) → Prop} {m : ℕ} [{ Γ := Dlt, rank := m }.Boldface P] (Γ : SigmaPiDelta) : { Γ := Γ, rank := m }.Boldface P

theorem LO.FirstOrder.Arith.HierarchySymbol.Boldface.of_sigma_of_pi {V : Type u_2} [ORingStruc V] {Γ : SigmaPiDelta} {k : ℕ} {P : (Fin k → V) → Prop} {m : ℕ} (hσ : { Γ := Sg, rank := m }.Boldface P) (hπ : { Γ := Pg, rank := m }.Boldface P) : { Γ := Γ, rank := m }.Boldface P

theorem LO.FirstOrder.Arith.HierarchySymbol.Boldface.of_zero {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} {k : ℕ} {P : (Fin k → V) → Prop} {Γ' : SigmaPiDelta} (h : { Γ := Γ', rank := 0 }.Boldface P) : ℌ.Boldface P

theorem LO.FirstOrder.Arith.HierarchySymbol.Boldface.of_deltaOne {V : Type u_2} [ORingStruc V] {k : ℕ} {P : (Fin k → V) → Prop} (h : Dlt1.Boldface P) {Γ : SigmaPiDelta} {m : ℕ} : { Γ := Γ, rank := m + 1 }.Boldface P

instance LO.FirstOrder.Arith.HierarchySymbol.Boldface.instOfSg0 {V : Type u_2} [ORingStruc V] {k : ℕ} {P : (Fin k → V) → Prop} [Sg0.Boldface P] (ℌ : HierarchySymbol) : ℌ.Boldface P

theorem LO.FirstOrder.Arith.HierarchySymbol.Boldface.retraction {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} {k : ℕ} {P : (Fin k → V) → Prop} (h : ℌ.Boldface P) {n : ℕ} (f : Fin k → Fin n) : ℌ.Boldface fun (v : Fin n → V) => P fun (i : Fin k) => v (f i)

theorem LO.FirstOrder.Arith.HierarchySymbol.Boldface.retractiont (n : ℕ) {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} {k : ℕ} {P : (Fin k → V) → Prop} (h : ℌ.Boldface P) (f : Fin k → Semiterm ℒₒᵣ V n) : ℌ.Boldface fun (v : Fin n → V) => P fun (i : Fin k) => Semiterm.valm V v id (f i)

@[simp]

theorem LO.FirstOrder.Arith.HierarchySymbol.Boldface.const {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} {k : ℕ} {P : Prop} : ℌ.Boldface fun (x : Fin k → V) => P

theorem LO.FirstOrder.Arith.HierarchySymbol.Boldface.and {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} {k : ℕ} {P Q : (Fin k → V) → Prop} (h₁ : ℌ.Boldface P) (h₂ : ℌ.Boldface Q) : ℌ.Boldface fun (v : Fin k → V) => P v ∧ Q v

theorem LO.FirstOrder.Arith.HierarchySymbol.Boldface.conj {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} {k l : ℕ} {P : Fin l → (Fin k → V) → Prop} (h : ∀ (i : Fin l), ℌ.Boldface fun (w : Fin k → V) => P i w) : ℌ.Boldface fun (v : Fin k → V) => ∀ (i : Fin l), P i v

theorem LO.FirstOrder.Arith.HierarchySymbol.Boldface.or {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} {k : ℕ} {P Q : (Fin k → V) → Prop} (h₁ : ℌ.Boldface P) (h₂ : ℌ.Boldface Q) : ℌ.Boldface fun (v : Fin k → V) => P v ∨ Q v

theorem LO.FirstOrder.Arith.HierarchySymbol.Boldface.not {V : Type u_2} [ORingStruc V] {Γ : SigmaPiDelta} {k : ℕ} {P : (Fin k → V) → Prop} {m : ℕ} (h : { Γ := Γ.alt, rank := m }.Boldface P) : { Γ := Γ, rank := m }.Boldface fun (v : Fin k → V) => ¬P v

theorem LO.FirstOrder.Arith.HierarchySymbol.Boldface.imp {V : Type u_2} [ORingStruc V] {Γ : SigmaPiDelta} {k : ℕ} {P Q : (Fin k → V) → Prop} {m : ℕ} (h₁ : { Γ := Γ.alt, rank := m }.Boldface P) (h₂ : { Γ := Γ, rank := m }.Boldface Q) : { Γ := Γ, rank := m }.Boldface fun (v : Fin k → V) => P v → Q v

theorem LO.FirstOrder.Arith.HierarchySymbol.Boldface.iff {V : Type u_2} [ORingStruc V] {k : ℕ} {P Q : (Fin k → V) → Prop} {m : ℕ} (h₁ : { Γ := Dlt, rank := m }.Boldface P) (h₂ : { Γ := Dlt, rank := m }.Boldface Q) {Γ : SigmaPiDelta} : { Γ := Γ, rank := m }.Boldface fun (v : Fin k → V) => P v ↔ Q v

theorem LO.FirstOrder.Arith.HierarchySymbol.Boldface.all {V : Type u_2} [ORingStruc V] {k s : ℕ} {P : (Fin k → V) → V → Prop} (h : { Γ := Pg, rank := s + 1 }.Boldface fun (w : Fin (k + 1) → V) => P (fun (x : Fin k) => w x.succ) (w 0)) : { Γ := Pg, rank := s + 1 }.Boldface fun (v : Fin k → V) => ∀ (x : V), P v x

theorem LO.FirstOrder.Arith.HierarchySymbol.Boldface.ex {V : Type u_2} [ORingStruc V] {k s : ℕ} {P : (Fin k → V) → V → Prop} (h : { Γ := Sg, rank := s + 1 }.Boldface fun (w : Fin (k + 1) → V) => P (fun (x : Fin k) => w x.succ) (w 0)) : { Γ := Sg, rank := s + 1 }.Boldface fun (v : Fin k → V) => ∃ (x : V), P v x

theorem LO.FirstOrder.Arith.HierarchySymbol.Boldface.equal' {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} {k : ℕ} (i j : Fin k) : ℌ.Boldface fun (v : Fin k → V) => v i = v j

theorem LO.FirstOrder.Arith.HierarchySymbol.Boldface.of_sigma {V : Type u_2} [ORingStruc V] {k m : ℕ} {f : (Fin k → V) → V} (h : { Γ := Sg, rank := m }.BoldfaceFunction f) {Γ : SigmaPiDelta} : { Γ := Γ, rank := m }.BoldfaceFunction f

theorem LO.FirstOrder.Arith.HierarchySymbol.Boldface.exVec {V : Type u_2} [ORingStruc V] {m k l : ℕ} {P : (Fin k → V) → (Fin l → V) → Prop} (h : { Γ := Sg, rank := m + 1 }.Boldface fun (w : Fin (k + l) → V) => P (fun (i : Fin k) => w (Fin.castAdd l i)) fun (j : Fin l) => w (Fin.natAdd k j)) : { Γ := Sg, rank := m + 1 }.Boldface fun (v : Fin k → V) => ∃ (ys : Fin l → V), P v ys

theorem LO.FirstOrder.Arith.HierarchySymbol.Boldface.allVec {V : Type u_2} [ORingStruc V] {m k l : ℕ} {P : (Fin k → V) → (Fin l → V) → Prop} (h : { Γ := Pg, rank := m + 1 }.Boldface fun (w : Fin (k + l) → V) => P (fun (i : Fin k) => w (Fin.castAdd l i)) fun (j : Fin l) => w (Fin.natAdd k j)) : { Γ := Pg, rank := m + 1 }.Boldface fun (v : Fin k → V) => ∀ (ys : Fin l → V), P v ys

theorem LO.FirstOrder.Arith.HierarchySymbol.Boldface.substitution {V : Type u_2} [ORingStruc V] {Γ : SigmaPiDelta} {k : ℕ} {P : (Fin k → V) → Prop} {l m : ℕ} {f : Fin k → (Fin l → V) → V} (hP : { Γ := Γ, rank := m + 1 }.Boldface P) (hf : ∀ (i : Fin k), { Γ := Sg, rank := m + 1 }.BoldfaceFunction (f i)) : { Γ := Γ, rank := m + 1 }.Boldface fun (z : Fin l → V) => P fun (i : Fin k) => f i z

theorem LO.FirstOrder.Arith.HierarchySymbol.BoldfacePred.comp {V : Type u_2} [ORingStruc V] {Γ : SigmaPiDelta} {m : ℕ} {P : V → Prop} {k : ℕ} {f : (Fin k → V) → V} (hP : { Γ := Γ, rank := m + 1 }-Predicate P) (hf : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f) : { Γ := Γ, rank := m + 1 }.Boldface fun (v : Fin k → V) => P (f v)

theorem LO.FirstOrder.Arith.HierarchySymbol.BoldfaceRel.comp {V : Type u_2} [ORingStruc V] {Γ : SigmaPiDelta} {m : ℕ} {P : V → V → Prop} {k : ℕ} {f g : (Fin k → V) → V} (hP : { Γ := Γ, rank := m + 1 }-Relation P) (hf : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f) (hg : { Γ := Sg, rank := m + 1 }.BoldfaceFunction g) : { Γ := Γ, rank := m + 1 }.Boldface fun (v : Fin k → V) => P (f v) (g v)

theorem LO.FirstOrder.Arith.HierarchySymbol.BoldfaceRel₃.comp {V : Type u_2} [ORingStruc V] {Γ : SigmaPiDelta} {m k : ℕ} {P : V → V → V → Prop} {f₁ f₂ f₃ : (Fin k → V) → V} (hP : { Γ := Γ, rank := m + 1 }-Relation₃ P) (hf₁ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f₁) (hf₂ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f₂) (hf₃ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f₃) : { Γ := Γ, rank := m + 1 }.Boldface fun (v : Fin k → V) => P (f₁ v) (f₂ v) (f₃ v)

theorem LO.FirstOrder.Arith.HierarchySymbol.BoldfaceRel₄.comp {V : Type u_2} [ORingStruc V] {Γ : SigmaPiDelta} {m k : ℕ} {P : V → V → V → V → Prop} {f₁ f₂ f₃ f₄ : (Fin k → V) → V} (hP : { Γ := Γ, rank := m + 1 }-Relation₄ P) (hf₁ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f₁) (hf₂ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f₂) (hf₃ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f₃) (hf₄ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f₄) : { Γ := Γ, rank := m + 1 }.Boldface fun (v : Fin k → V) => P (f₁ v) (f₂ v) (f₃ v) (f₄ v)

theorem LO.FirstOrder.Arith.HierarchySymbol.BoldfaceRel₅.comp {V : Type u_2} [ORingStruc V] {Γ : SigmaPiDelta} {m k : ℕ} {P : V → V → V → V → V → Prop} {f₁ f₂ f₃ f₄ f₅ : (Fin k → V) → V} (hP : { Γ := Γ, rank := m + 1 }-Relation₅ P) (hf₁ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f₁) (hf₂ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f₂) (hf₃ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f₃) (hf₄ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f₄) (hf₅ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f₅) : { Γ := Γ, rank := m + 1 }.Boldface fun (v : Fin k → V) => P (f₁ v) (f₂ v) (f₃ v) (f₄ v) (f₅ v)

theorem LO.FirstOrder.Arith.HierarchySymbol.Boldface.comp₁ {V : Type u_2} [ORingStruc V] {Γ : SigmaPiDelta} {m k : ℕ} {P : V → Prop} {f : (Fin k → V) → V} [{ Γ := Γ, rank := m + 1 }-Predicate P] (hf : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f) : { Γ := Γ, rank := m + 1 }.Boldface fun (v : Fin k → V) => P (f v)

theorem LO.FirstOrder.Arith.HierarchySymbol.Boldface.comp₂ {V : Type u_2} [ORingStruc V] {Γ : SigmaPiDelta} {m k : ℕ} {P : V → V → Prop} {f g : (Fin k → V) → V} [{ Γ := Γ, rank := m + 1 }-Relation P] (hf : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f) (hg : { Γ := Sg, rank := m + 1 }.BoldfaceFunction g) : { Γ := Γ, rank := m + 1 }.Boldface fun (v : Fin k → V) => P (f v) (g v)

theorem LO.FirstOrder.Arith.HierarchySymbol.Boldface.comp₃ {V : Type u_2} [ORingStruc V] {Γ : SigmaPiDelta} {m k : ℕ} {P : V → V → V → Prop} {f₁ f₂ f₃ : (Fin k → V) → V} [{ Γ := Γ, rank := m + 1 }-Relation₃ P] (hf₁ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f₁) (hf₂ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f₂) (hf₃ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f₃) : { Γ := Γ, rank := m + 1 }.Boldface fun (v : Fin k → V) => P (f₁ v) (f₂ v) (f₃ v)

theorem LO.FirstOrder.Arith.HierarchySymbol.Boldface.comp₄ {V : Type u_2} [ORingStruc V] {Γ : SigmaPiDelta} {m k : ℕ} {P : V → V → V → V → Prop} {f₁ f₂ f₃ f₄ : (Fin k → V) → V} [{ Γ := Γ, rank := m + 1 }-Relation₄ P] (hf₁ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f₁) (hf₂ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f₂) (hf₃ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f₃) (hf₄ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f₄) : { Γ := Γ, rank := m + 1 }.Boldface fun (v : Fin k → V) => P (f₁ v) (f₂ v) (f₃ v) (f₄ v)

theorem LO.FirstOrder.Arith.HierarchySymbol.Boldface.comp₅ {V : Type u_2} [ORingStruc V] {Γ : SigmaPiDelta} {m k : ℕ} {P : V → V → V → V → V → Prop} {f₁ f₂ f₃ f₄ f₅ : (Fin k → V) → V} [{ Γ := Γ, rank := m + 1 }-Relation₅ P] (hf₁ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f₁) (hf₂ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f₂) (hf₃ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f₃) (hf₄ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f₄) (hf₅ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f₅) : { Γ := Γ, rank := m + 1 }.Boldface fun (v : Fin k → V) => P (f₁ v) (f₂ v) (f₃ v) (f₄ v) (f₅ v)

theorem LO.FirstOrder.Arith.HierarchySymbol.BoldfacePred.of_iff {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} {P Q : V → Prop} (H : ℌ-Predicate Q) (h : ∀ (x : V), P x ↔ Q x) : ℌ-Predicate P

instance LO.FirstOrder.Arith.HierarchySymbol.BoldfaceFunction₁.graph {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} {f : V → V} [h : ℌ-Function₁ f] : ℌ-Relation Function.Graph f

instance LO.FirstOrder.Arith.HierarchySymbol.BoldfaceFunction₂.graph {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} {f : V → V → V} [h : ℌ-Function₂ f] : ℌ-Relation₃ Function.Graph₂ f

instance LO.FirstOrder.Arith.HierarchySymbol.BoldfaceFunction₃.graph {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} {f : V → V → V → V} [h : ℌ-Function₃ f] : ℌ-Relation₄ Function.Graph₃ f

theorem LO.FirstOrder.Arith.HierarchySymbol.BoldfaceFunction.graph_delta {V : Type u_2} [ORingStruc V] {m k : ℕ} {f : (Fin k → V) → V} (h : { Γ := Sg, rank := m }.BoldfaceFunction f) : { Γ := Dlt, rank := m }.BoldfaceFunction f

instance LO.FirstOrder.Arith.HierarchySymbol.BoldfaceFunction.instMkDeltaSigmaPiDeltaOfSigma {V : Type u_2} [ORingStruc V] {m k : ℕ} {f : (Fin k → V) → V} [h : { Γ := Sg, rank := m }.BoldfaceFunction f] : { Γ := Dlt, rank := m }.BoldfaceFunction f

instance LO.FirstOrder.Arith.HierarchySymbol.BoldfaceFunction.instOfSg0 {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} {k : ℕ} {f : (Fin k → V) → V} [Sg0.BoldfaceFunction f] : ℌ.BoldfaceFunction f

theorem LO.FirstOrder.Arith.HierarchySymbol.BoldfaceFunction.of_sigmaOne {V : Type u_2} [ORingStruc V] {k : ℕ} {f : (Fin k → V) → V} (h : Sg1.BoldfaceFunction f) {Γ : SigmaPiDelta} {m : ℕ} : { Γ := Γ, rank := m + 1 }.BoldfaceFunction f

@[simp]

theorem LO.FirstOrder.Arith.HierarchySymbol.BoldfaceFunction.var {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} {k : ℕ} (i : Fin k) : ℌ.BoldfaceFunction fun (v : Fin k → V) => v i

@[simp]

theorem LO.FirstOrder.Arith.HierarchySymbol.BoldfaceFunction.const {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} {k : ℕ} (c : V) : ℌ.BoldfaceFunction fun (x : Fin k → V) => c

@[simp]

theorem LO.FirstOrder.Arith.HierarchySymbol.BoldfaceFunction.term_retraction (n : ℕ) {V : Type u_2} [ORingStruc V] {k : ℕ} {ℌ : HierarchySymbol} (t : Semiterm ℒₒᵣ V n) (e : Fin n → Fin k) : ℌ.BoldfaceFunction fun (v : Fin k → V) => Semiterm.valm V (fun (x : Fin n) => v (e x)) id t

@[simp]

theorem LO.FirstOrder.Arith.HierarchySymbol.BoldfaceFunction.term {V : Type u_2} [ORingStruc V] {k : ℕ} {ℌ : HierarchySymbol} (t : Semiterm ℒₒᵣ V k) : ℌ.BoldfaceFunction fun (v : Fin k → V) => Semiterm.valm V v id t

theorem LO.FirstOrder.Arith.HierarchySymbol.BoldfaceFunction.of_eq {V : Type u_2} [ORingStruc V] {k : ℕ} {ℌ : HierarchySymbol} {f : (Fin k → V) → V} (g : (Fin k → V) → V) (h : ∀ (v : Fin k → V), f v = g v) (H : ℌ.BoldfaceFunction f) : ℌ.BoldfaceFunction g

theorem LO.FirstOrder.Arith.HierarchySymbol.BoldfaceFunction.retraction {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} {n k : ℕ} {f : (Fin k → V) → V} (hf : ℌ.BoldfaceFunction f) (e : Fin k → Fin n) : ℌ.BoldfaceFunction fun (v : Fin n → V) => f fun (i : Fin k) => v (e i)

theorem LO.FirstOrder.Arith.HierarchySymbol.BoldfaceFunction.retractiont {V : Type u_2} [ORingStruc V] {ℌ : HierarchySymbol} {n k : ℕ} {f : (Fin k → V) → V} (hf : ℌ.BoldfaceFunction f) (t : Fin k → Semiterm ℒₒᵣ V n) : ℌ.BoldfaceFunction fun (v : Fin n → V) => f fun (i : Fin k) => Semiterm.valm V v id (t i)

theorem LO.FirstOrder.Arith.HierarchySymbol.BoldfaceFunction.rel {V : Type u_2} [ORingStruc V] {k : ℕ} {ℌ : HierarchySymbol} {f : (Fin k → V) → V} (h : ℌ.BoldfaceFunction f) : ℌ.Boldface fun (v : Fin (k + 1) → V) => v 0 = f fun (x : Fin k) => v x.succ

theorem LO.FirstOrder.Arith.HierarchySymbol.BoldfaceFunction.nth {V : Type u_2} [ORingStruc V] {k : ℕ} (ℌ : HierarchySymbol) (i : Fin k) : ℌ.BoldfaceFunction fun (w : Fin k → V) => w i

theorem LO.FirstOrder.Arith.HierarchySymbol.BoldfaceFunction.substitution {V : Type u_2} [ORingStruc V] {Γ : SigmaPiDelta} {k l m : ℕ} {F : (Fin k → V) → V} {f : Fin k → (Fin l → V) → V} (hF : { Γ := Γ, rank := m + 1 }.BoldfaceFunction F) (hf : ∀ (i : Fin k), { Γ := Sg, rank := m + 1 }.BoldfaceFunction (f i)) : { Γ := Γ, rank := m + 1 }.BoldfaceFunction fun (z : Fin l → V) => F fun (i : Fin k) => f i z

theorem LO.FirstOrder.Arith.HierarchySymbol.BoldfaceFunction₁.comp {V : Type u_2} [ORingStruc V] {Γ : SigmaPiDelta} {m k : ℕ} {F : V → V} {f : (Fin k → V) → V} (hF : { Γ := Γ, rank := m + 1 }-Function₁ F) (hf : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f) : { Γ := Γ, rank := m + 1 }.BoldfaceFunction fun (v : Fin k → V) => F (f v)

theorem LO.FirstOrder.Arith.HierarchySymbol.BoldfaceFunction₂.comp {V : Type u_2} [ORingStruc V] {Γ : SigmaPiDelta} {m k : ℕ} {F : V → V → V} {f₁ f₂ : (Fin k → V) → V} (hF : { Γ := Γ, rank := m + 1 }-Function₂ F) (hf₁ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f₁) (hf₂ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f₂) : { Γ := Γ, rank := m + 1 }.BoldfaceFunction fun (v : Fin k → V) => F (f₁ v) (f₂ v)

theorem LO.FirstOrder.Arith.HierarchySymbol.BoldfaceFunction₃.comp {V : Type u_2} [ORingStruc V] {Γ : SigmaPiDelta} {m k : ℕ} {F : V → V → V → V} {f₁ f₂ f₃ : (Fin k → V) → V} (hF : { Γ := Γ, rank := m + 1 }-Function₃ F) (hf₁ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f₁) (hf₂ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f₂) (hf₃ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f₃) : { Γ := Γ, rank := m + 1 }.BoldfaceFunction fun (v : Fin k → V) => F (f₁ v) (f₂ v) (f₃ v)

theorem LO.FirstOrder.Arith.HierarchySymbol.BoldfaceFunction₄.comp {V : Type u_2} [ORingStruc V] {Γ : SigmaPiDelta} {m k : ℕ} {F : V → V → V → V → V} {f₁ f₂ f₃ f₄ : (Fin k → V) → V} (hF : { Γ := Γ, rank := m + 1 }-Function₄ F) (hf₁ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f₁) (hf₂ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f₂) (hf₃ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f₃) (hf₄ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f₄) : { Γ := Γ, rank := m + 1 }.BoldfaceFunction fun (v : Fin k → V) => F (f₁ v) (f₂ v) (f₃ v) (f₄ v)

theorem LO.FirstOrder.Arith.HierarchySymbol.BoldfaceFunction₅.comp {V : Type u_2} [ORingStruc V] {Γ : SigmaPiDelta} {m k : ℕ} {F : V → V → V → V → V → V} {f₁ f₂ f₃ f₄ f₅ : (Fin k → V) → V} (hF : { Γ := Γ, rank := m + 1 }.BoldfaceFunction₅ F) (hf₁ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f₁) (hf₂ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f₂) (hf₃ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f₃) (hf₄ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f₄) (hf₅ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f₅) : { Γ := Γ, rank := m + 1 }.BoldfaceFunction fun (v : Fin k → V) => F (f₁ v) (f₂ v) (f₃ v) (f₄ v) (f₅ v)

theorem LO.FirstOrder.Arith.HierarchySymbol.BoldfaceFunction.comp₁ {V : Type u_2} [ORingStruc V] {Γ : SigmaPiDelta} {m k : ℕ} {f : V → V} [{ Γ := Γ, rank := m + 1 }-Function₁ f] {g : (Fin k → V) → V} (hg : { Γ := Sg, rank := m + 1 }.BoldfaceFunction g) : { Γ := Γ, rank := m + 1 }.BoldfaceFunction fun (v : Fin k → V) => f (g v)

theorem LO.FirstOrder.Arith.HierarchySymbol.BoldfaceFunction.comp₂ {V : Type u_2} [ORingStruc V] {Γ : SigmaPiDelta} {m k : ℕ} {f : V → V → V} [{ Γ := Γ, rank := m + 1 }-Function₂ f] {g₁ g₂ : (Fin k → V) → V} (hg₁ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction g₁) (hg₂ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction g₂) : { Γ := Γ, rank := m + 1 }.BoldfaceFunction fun (v : Fin k → V) => f (g₁ v) (g₂ v)

theorem LO.FirstOrder.Arith.HierarchySymbol.BoldfaceFunction.comp₃ {V : Type u_2} [ORingStruc V] {Γ : SigmaPiDelta} {m k : ℕ} {f : V → V → V → V} [{ Γ := Γ, rank := m + 1 }-Function₃ f] {g₁ g₂ g₃ : (Fin k → V) → V} (hg₁ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction g₁) (hg₂ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction g₂) (hg₃ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction g₃) : { Γ := Γ, rank := m + 1 }.BoldfaceFunction fun (v : Fin k → V) => f (g₁ v) (g₂ v) (g₃ v)

theorem LO.FirstOrder.Arith.HierarchySymbol.BoldfaceFunction.comp₄ {V : Type u_2} [ORingStruc V] {Γ : SigmaPiDelta} {m k : ℕ} {f : V → V → V → V → V} [{ Γ := Γ, rank := m + 1 }-Function₄ f] {g₁ g₂ g₃ g₄ : (Fin k → V) → V} (hg₁ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction g₁) (hg₂ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction g₂) (hg₃ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction g₃) (hg₄ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction g₄) : { Γ := Γ, rank := m + 1 }.BoldfaceFunction fun (v : Fin k → V) => f (g₁ v) (g₂ v) (g₃ v) (g₄ v)

theorem LO.FirstOrder.Arith.HierarchySymbol.BoldfaceFunction.comp₅ {V : Type u_2} [ORingStruc V] {Γ : SigmaPiDelta} {m k : ℕ} {f : V → V → V → V → V → V} [{ Γ := Γ, rank := m + 1 }.BoldfaceFunction₅ f] {g₁ g₂ g₃ g₄ g₅ : (Fin k → V) → V} (hg₁ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction g₁) (hg₂ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction g₂) (hg₃ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction g₃) (hg₄ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction g₄) (hg₅ : { Γ := Sg, rank := m + 1 }.BoldfaceFunction g₅) : { Γ := Γ, rank := m + 1 }.BoldfaceFunction fun (v : Fin k → V) => f (g₁ v) (g₂ v) (g₃ v) (g₄ v) (g₅ v)

theorem LO.FirstOrder.Arith.HierarchySymbol.Boldface.ball_lt {V : Type u_2} [ORingStruc V] {k m : ℕ} {Γ : SigmaPiDelta} {P : (Fin k → V) → V → Prop} {f : (Fin k → V) → V} (hf : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f) (h : { Γ := Γ, rank := m + 1 }.Boldface fun (w : Fin (k + 1) → V) => P (fun (x : Fin k) => w x.succ) (w 0)) : { Γ := Γ, rank := m + 1 }.Boldface fun (v : Fin k → V) => ∀ x < f v, P v x

theorem LO.FirstOrder.Arith.HierarchySymbol.Boldface.bex_lt {V : Type u_2} [ORingStruc V] {k m : ℕ} {Γ : SigmaPiDelta} {P : (Fin k → V) → V → Prop} {f : (Fin k → V) → V} (hf : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f) (h : { Γ := Γ, rank := m + 1 }.Boldface fun (w : Fin (k + 1) → V) => P (fun (x : Fin k) => w x.succ) (w 0)) : { Γ := Γ, rank := m + 1 }.Boldface fun (v : Fin k → V) => ∃ x < f v, P v x

theorem LO.FirstOrder.Arith.HierarchySymbol.Boldface.ball_le {V : Type u_2} [ORingStruc V] {k m : ℕ} [V ⊧ₘ* 𝐏𝐀⁻] {Γ : SigmaPiDelta} {P : (Fin k → V) → V → Prop} {f : (Fin k → V) → V} (hf : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f) (h : { Γ := Γ, rank := m + 1 }.Boldface fun (w : Fin (k + 1) → V) => P (fun (x : Fin k) => w x.succ) (w 0)) : { Γ := Γ, rank := m + 1 }.Boldface fun (v : Fin k → V) => ∀ x ≤ f v, P v x

theorem LO.FirstOrder.Arith.HierarchySymbol.Boldface.bex_le {V : Type u_2} [ORingStruc V] {k m : ℕ} [V ⊧ₘ* 𝐏𝐀⁻] {Γ : SigmaPiDelta} {P : (Fin k → V) → V → Prop} {f : (Fin k → V) → V} (hf : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f) (h : { Γ := Γ, rank := m + 1 }.Boldface fun (w : Fin (k + 1) → V) => P (fun (x : Fin k) => w x.succ) (w 0)) : { Γ := Γ, rank := m + 1 }.Boldface fun (v : Fin k → V) => ∃ x ≤ f v, P v x

theorem LO.FirstOrder.Arith.HierarchySymbol.Boldface.ball_lt' {V : Type u_2} [ORingStruc V] {k m : ℕ} {Γ : SigmaPiDelta} {P : (Fin k → V) → V → Prop} {f : (Fin k → V) → V} (hf : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f) (h : { Γ := Γ, rank := m + 1 }.Boldface fun (w : Fin (k + 1) → V) => P (fun (x : Fin k) => w x.succ) (w 0)) : { Γ := Γ, rank := m + 1 }.Boldface fun (v : Fin k → V) => ∀ {x : V}, x < f v → P v x

theorem LO.FirstOrder.Arith.HierarchySymbol.Boldface.ball_le' {V : Type u_2} [ORingStruc V] {k m : ℕ} [V ⊧ₘ* 𝐏𝐀⁻] {Γ : SigmaPiDelta} {P : (Fin k → V) → V → Prop} {f : (Fin k → V) → V} (hf : { Γ := Sg, rank := m + 1 }.BoldfaceFunction f) (h : { Γ := Γ, rank := m + 1 }.Boldface fun (w : Fin (k + 1) → V) => P (fun (x : Fin k) => w x.succ) (w 0)) : { Γ := Γ, rank := m + 1 }.Boldface fun (v : Fin k → V) => ∀ {x : V}, x ≤ f v → P v x