Foundation.IntFO.Translation

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instance LO.FirstOrder.Sequent.instTildeListSemiformula {L : Language} {ξ : Type u_2} {n : ℕ} :

Tilde (List (Semiformula L ξ n))

Equations

LO.FirstOrder.Sequent.instTildeListSemiformula = { tilde := fun (Γ : List (LO.FirstOrder.Semiformula L ξ n)) => List.map (fun (x : LO.FirstOrder.Semiformula L ξ n) => ∼x) Γ }

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@[simp]

theorem LO.FirstOrder.Sequent.neg_def {L : Language} {ξ : Type u_2} {n : ℕ} (Γ : List (Semiformula L ξ n)) :

∼Γ = List.map (fun (x : Semiformula L ξ n) => ∼x) Γ

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@[simp]

theorem LO.FirstOrder.Sequent.neg_nil {L : Language} {ξ : Type u_2} {n : ℕ} :

∼[] = []

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@[simp]

theorem LO.FirstOrder.Sequent.neg_cons {L : Language} {ξ : Type u_2} {n : ℕ} (Γ : List (Semiformula L ξ n)) (φ : Semiformula L ξ n) :

∼(φ :: Γ) = ∼φ :: ∼Γ

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@[simp]

theorem LO.FirstOrder.Sequent.neg_neg_eq {L : Language} {ξ : Type u_2} {n : ℕ} (Γ : List (Semiformula L ξ n)) :

∼∼Γ = Γ

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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 = ∀' φ.doubleNegation
φ.ex.doubleNegation = ∼(∀' ∼φ.doubleNegation)

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def LO.FirstOrder.«term_ᴺ» :

Lean.TrailingParserDescr

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

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@[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

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@[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

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@[simp]

theorem LO.FirstOrder.Semiformula.doubleNegation_verum {L : Language} {ξ : Type u_2} {n : ℕ} :

⊤.doubleNegation = ∼⊥

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@[simp]

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

⊥.doubleNegation = ⊥

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@[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

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@[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)

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@[simp]

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

(∀' φ).doubleNegation = ∀' φ.doubleNegation

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@[simp]

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

(∃' φ).doubleNegation = ∼(∀' ∼φ.doubleNegation)

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@[simp]

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

φ.doubleNegation.IsNegative

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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

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theorem LO.FirstOrder.Semiformula.substitute_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

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@[reducible, inline]

abbrev LO.FirstOrder.Theory.doubleNegation {L : Language} (T : Theory L) :

Theoryᵢ L

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T.doubleNegation = LO.FirstOrder.Semiformula.doubleNegation '' T

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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")

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@[reducible, inline]

abbrev LO.FirstOrder.Sequent.doubleNegation {L : Language} {ξ : Type u_2} {n : ℕ} (Γ : List (Semiformula L ξ n)) :

List (Semiformulaᵢ L ξ n)

Equations

LO.FirstOrder.Sequent.doubleNegation Γ = List.map (fun (x : LO.FirstOrder.Semiformula L ξ n) => x.doubleNegation) Γ

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def LO.FirstOrder.«term_ᴺ_2» :

Lean.TrailingParserDescr

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LO.FirstOrder.«term_ᴺ_2» = Lean.ParserDescr.trailingNode `LO.FirstOrder.«term_ᴺ_2» 1024 1024 (Lean.ParserDescr.symbol "ᴺ")

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@[irreducible]

noncomputable def LO.FirstOrder.Derivation.negDoubleNegation {L : Language} [L.DecidableEq] (φ : SyntacticFormula L) :

𝐌𝐢𝐧¹ ⊢ ∼Semiformula.doubleNegation φ ⭤ (∼φ).doubleNegation

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noncomputable def LO.FirstOrder.Derivation.goedelGentzen {L : Language} [L.DecidableEq] {Γ : Sequent L} :

⊢ᵀ Γ → Sequent.doubleNegation (∼Γ) ⊢[𝐌𝐢𝐧¹ ] ⊥

Equations

One or more equations did not get rendered due to their size.
LO.FirstOrder.Derivation.goedelGentzen (LO.FirstOrder.Derivation.axL Γ_2 r v) = LO.System.FiniteContext.nthAxm 1 ⋯⨀LO.System.FiniteContext.nthAxm 0 ⋯
LO.FirstOrder.Derivation.goedelGentzen (LO.FirstOrder.Derivation.verum Γ_2) = LO.System.FiniteContext.nthAxm 0 ⋯
LO.FirstOrder.Derivation.goedelGentzen (d.wk h) = LO.System.FiniteContext.weakening ⋯ (LO.FirstOrder.Derivation.goedelGentzen d)