def LO.FirstOrder.Semiformula.doubleNegation {L : Language} {ξ : Type u_2} {n : ℕ} : Semiformula L ξ n → Semiformulaᵢ L ξ n

Equations

• (LO.FirstOrder.Semiformula.rel r v).doubleNegation = ∼∼LO.FirstOrder.Semiformulaᵢ.rel r v
• (LO.FirstOrder.Semiformula.nrel r v).doubleNegation = ∼LO.FirstOrder.Semiformulaᵢ.rel r v
• LO.FirstOrder.Semiformula.verum.doubleNegation = ⊤
• LO.FirstOrder.Semiformula.falsum.doubleNegation = ⊥
• (φ.and ψ).doubleNegation = φ.doubleNegation ⋏ ψ.doubleNegation
• (φ.or ψ).doubleNegation = ∼(∼φ.doubleNegation ⋏ ∼ψ.doubleNegation)
• φ.all.doubleNegation = ∀¹ φ.doubleNegation
• φ.exs.doubleNegation = ∼(∀¹ ∼φ.doubleNegation)

def LO.FirstOrder.«term_ᴺ» : Lean.TrailingParserDescr

Equations

• LO.FirstOrder.«term_ᴺ» = Lean.ParserDescr.trailingNode `LO.FirstOrder.«term_ᴺ» 1024 1024 (Lean.ParserDescr.symbol "ᴺ")

@[simp]

theorem LO.FirstOrder.Semiformula.doubleNegation_rel {L : Language} {ξ : Type u_2} {n k : ℕ} (r : L.Rel k) (v : Fin k → Semiterm L ξ n) : (rel r v).doubleNegation = ∼∼Semiformulaᵢ.rel r v

@[simp]

theorem LO.FirstOrder.Semiformula.doubleNegation_nrel {L : Language} {ξ : Type u_2} {n k : ℕ} (r : L.Rel k) (v : Fin k → Semiterm L ξ n) : (nrel r v).doubleNegation = ∼Semiformulaᵢ.rel r v

@[simp]

theorem LO.FirstOrder.Semiformula.doubleNegation_verum {L : Language} {ξ : Type u_2} {n : ℕ} : ⊤.doubleNegation = ∼⊥

@[simp]

theorem LO.FirstOrder.Semiformula.doubleNegation_falsum {L : Language} {ξ : Type u_2} {n : ℕ} : ⊥.doubleNegation = ⊥

@[simp]

theorem LO.FirstOrder.Semiformula.doubleNegation_and {L : Language} {ξ : Type u_2} {n : ℕ} (φ ψ : Semiformula L ξ n) : (φ ⋏ ψ).doubleNegation = φ.doubleNegation ⋏ ψ.doubleNegation

@[simp]

theorem LO.FirstOrder.Semiformula.doubleNegation_or {L : Language} {ξ : Type u_2} {n : ℕ} (φ ψ : Semiformula L ξ n) : (φ ⋎ ψ).doubleNegation = ∼(∼φ.doubleNegation ⋏ ∼ψ.doubleNegation)

@[simp]

theorem LO.FirstOrder.Semiformula.doubleNegation_all {L : Language} {ξ : Type u_2} {n : ℕ} (φ : Semiformula L ξ (n + 1)) : (∀¹ φ).doubleNegation = ∀¹ φ.doubleNegation

@[simp]

theorem LO.FirstOrder.Semiformula.doubleNegation_ex {L : Language} {ξ : Type u_2} {n : ℕ} (φ : Semiformula L ξ (n + 1)) : (∃¹ φ).doubleNegation = ∼(∀¹ ∼φ.doubleNegation)

theorem LO.FirstOrder.Semiformula.doubleNegation_imply {L : Language} {ξ : Type u_2} {n : ℕ} (φ ψ : Semiformula L ξ n) : (φ 🡒 ψ).doubleNegation = ∼(∼(∼φ).doubleNegation ⋏ ∼ψ.doubleNegation)

@[simp]

theorem LO.FirstOrder.Semiformula.doubleNegation_isNegative {L : Language} {ξ : Type u_2} {n : ℕ} (φ : Semiformula L ξ n) : φ.doubleNegation.IsNegative

@[simp]

theorem LO.FirstOrder.Semiformula.doubleNegation_conj₂ {L : Language} {ξ : Type u_2} {n : ℕ} (Γ : List (Semiformula L ξ n)) : (⋀Γ).doubleNegation = ⋀List.map doubleNegation Γ

theorem LO.FirstOrder.Semiformula.doubleNegation_fconj {L : Language} {ξ : Type u_2} {n : ℕ} (s : Finset (Semiformula L ξ n)) : s.conj.doubleNegation = ⋀List.map doubleNegation s.toList

theorem LO.FirstOrder.Semiformula.rew_doubleNegation {L : Language} {ξ₁ : Type u_2} {n₁ : ℕ} {ξ₂ : Type u_3} {n₂ : ℕ} (ω : Rew L ξ₁ n₁ ξ₂ n₂) (φ : Semiformula L ξ₁ n₁) : (Rewriting.app ω) φ.doubleNegation = ((Rewriting.app ω) φ).doubleNegation

theorem LO.FirstOrder.Semiformula.subst_doubleNegation {L : Language} {ξ : Type u_2} {n₁ n₂ : ℕ} (φ : Semiformula L ξ n₁) (v : Fin n₁ → Semiterm L ξ n₂) : φ.doubleNegation ⇜ v = (φ ⇜ v).doubleNegation

theorem LO.FirstOrder.Semiformula.emb_doubleNegation {L : Language} {n₁ : ℕ} {ξ : Type u_2} (φ : Semisentence L n₁) : Rewriting.emb (doubleNegation φ) = (Rewriting.emb φ).doubleNegation

def LO.FirstOrder.Sequent.doubleNegation {L : Language} (Γ : Sequent L) : Sequentᵢ L

Equations

• Γ.doubleNegation = List.map LO.FirstOrder.Semiformula.doubleNegation Γ

def LO.FirstOrder.«term_ᴺ_1» : Lean.TrailingParserDescr

Equations

• LO.FirstOrder.«term_ᴺ_1» = Lean.ParserDescr.trailingNode `LO.FirstOrder.«term_ᴺ_1» 1024 1024 (Lean.ParserDescr.symbol "ᴺ")

@[simp]

theorem LO.FirstOrder.Sequent.doubleNegation_nil {L : Language} : doubleNegation [] = []

@[simp]

theorem LO.FirstOrder.Sequent.doubleNegation_cons {L : Language} (φ : Proposition L) (Γ : Sequent L) : doubleNegation (φ :: Γ) = Semiformula.doubleNegation φ :: Γ.doubleNegation

@[simp]

theorem LO.FirstOrder.Sequent.doubleNegation_append {L : Language} (Γ Δ : Sequent L) : (Γ ++ Δ).doubleNegation = Γ.doubleNegation ++ Δ.doubleNegation

theorem LO.FirstOrder.Sequent.shift_doubleNegation {L : Language} (Γ : Sequent L) : Rewriting.shifts Γ.doubleNegation = doubleNegation (Rewriting.shifts Γ)

def LO.FirstOrder.Theory.ToTheoryᵢ {L : Language} (T : Theory L) (Λ : Hilbertᵢ L) : Theoryᵢ L Λ

Equations

• T.ToTheoryᵢ Λ = { theory := LO.FirstOrder.Semiformula.doubleNegation '' T }

@[simp]

theorem LO.FirstOrder.Theory.ToTheoryᵢ_theory_def {L : Language} (T : Theory L) (Λ : Hilbertᵢ L) : (T.ToTheoryᵢ Λ).theory = Semiformula.doubleNegation '' T

@[irreducible]

def LO.FirstOrder.Derivation.negDoubleNegation {L : Language} [L.DecidableEq] {Λ : Hilbertᵢ L} (φ : Proposition L) : Λ ⊢! ∼Semiformula.doubleNegation φ 🡘 (∼φ).doubleNegation

Equations

• One or more equations did not get rendered due to their size.
• LO.FirstOrder.Derivation.negDoubleNegation (LO.FirstOrder.Semiformula.rel r v) = LO.Entailment.tneIff
• LO.FirstOrder.Derivation.negDoubleNegation (LO.FirstOrder.Semiformula.nrel r v) = LO.Entailment.E_Id
• LO.FirstOrder.Derivation.negDoubleNegation LO.FirstOrder.Semiformula.verum = LO.Entailment.ENNOO
• LO.FirstOrder.Derivation.negDoubleNegation LO.FirstOrder.Semiformula.falsum = LO.Entailment.E_Id

theorem LO.FirstOrder.Derivation.neg_doubleNegation {L : Language} [L.DecidableEq] {Λ : Hilbertᵢ L} (φ : Proposition L) : Λ ⊢ ∼Semiformula.doubleNegation φ 🡘 (∼φ).doubleNegation

theorem LO.FirstOrder.Derivation.neg_doubleNegation' {L : Language} [L.DecidableEq] {Λ : Hilbertᵢ L} (φ : Proposition L) : Λ ⊢ ∼(∼φ).doubleNegation 🡘 Semiformula.doubleNegation φ

theorem LO.FirstOrder.Derivation.imply_doubleNegation {L : Language} [L.DecidableEq] {Λ : Hilbertᵢ L} (φ ψ : Proposition L) : Λ ⊢ (Semiformula.doubleNegation φ 🡒 Semiformula.doubleNegation ψ) 🡘 (φ 🡒 ψ).doubleNegation

def LO.FirstOrder.Derivation.gödelGentzen {L : Language} [L.DecidableEq] {Λ : Hilbertᵢ L} {Γ : Sequent L} : ⊢ᴸᴷ¹ Γ → (∼Γ).doubleNegation ⊢[Λ]! ⊥

Equations

• One or more equations did not get rendered due to their size.
• (LO.FirstOrder.Derivation.identity r v).gödelGentzen = LO.Entailment.FiniteContext.nthAxm 1 ⋯⨀!LO.Entailment.FiniteContext.nthAxm 0 ⋯
• LO.FirstOrder.Derivation.verum.gödelGentzen = LO.Entailment.FiniteContext.nthAxm 0 ⋯
• (d.contraction h).gödelGentzen = LO.Entailment.FiniteContext.weakening ⋯ d.gödelGentzen

theorem LO.FirstOrder.Provable.gödel_gentzen {L : Language} {φ : Proposition L} {Λ : Hilbertᵢ L} : 𝐋𝐊¹ ⊢ φ → Λ ⊢ Semiformula.doubleNegation φ