Derivation2

Different characterizations of proof.

inductive LO.FirstOrder.Derivation2 {L : LO.FirstOrder.Language} [(k : ℕ) → DecidableEq (L.Func k)] [(k : ℕ) → DecidableEq (L.Rel k)] (T : LO.FirstOrder.Theory L) : Finset (LO.FirstOrder.SyntacticFormula L) → Type u_1

• closed: {L : LO.FirstOrder.Language} → [inst : (k : ℕ) → DecidableEq (L.Func k)] → [inst_1 : (k : ℕ) → DecidableEq (L.Rel k)] → {T : LO.FirstOrder.Theory L} → (Δ : Finset (LO.FirstOrder.SyntacticFormula L)) → (φ : LO.FirstOrder.SyntacticFormula L) → φ ∈ Δ → ∼φ ∈ Δ → LO.FirstOrder.Derivation2 T Δ
• root: {L : LO.FirstOrder.Language} → [inst : (k : ℕ) → DecidableEq (L.Func k)] → [inst_1 : (k : ℕ) → DecidableEq (L.Rel k)] → {T : LO.FirstOrder.Theory L} → {Δ : Finset (LO.FirstOrder.SyntacticFormula L)} → (φ : LO.FirstOrder.SyntacticFormula L) → φ ∈ T → φ ∈ Δ → LO.FirstOrder.Derivation2 T Δ
• verum: {L : LO.FirstOrder.Language} → [inst : (k : ℕ) → DecidableEq (L.Func k)] → [inst_1 : (k : ℕ) → DecidableEq (L.Rel k)] → {T : LO.FirstOrder.Theory L} → {Δ : Finset (LO.FirstOrder.SyntacticFormula L)} → ⊤ ∈ Δ → LO.FirstOrder.Derivation2 T Δ
• and: {L : LO.FirstOrder.Language} → [inst : (k : ℕ) → DecidableEq (L.Func k)] → [inst_1 : (k : ℕ) → DecidableEq (L.Rel k)] → {T : LO.FirstOrder.Theory L} → {Δ : Finset (LO.FirstOrder.SyntacticFormula L)} → {φ ψ : LO.FirstOrder.SyntacticFormula L} → φ ⋏ ψ ∈ Δ → LO.FirstOrder.Derivation2 T (insert φ Δ) → LO.FirstOrder.Derivation2 T (insert ψ Δ) → LO.FirstOrder.Derivation2 T Δ
• or: {L : LO.FirstOrder.Language} → [inst : (k : ℕ) → DecidableEq (L.Func k)] → [inst_1 : (k : ℕ) → DecidableEq (L.Rel k)] → {T : LO.FirstOrder.Theory L} → {Δ : Finset (LO.FirstOrder.SyntacticFormula L)} → {φ ψ : LO.FirstOrder.SyntacticFormula L} → φ ⋎ ψ ∈ Δ → LO.FirstOrder.Derivation2 T (insert φ (insert ψ Δ)) → LO.FirstOrder.Derivation2 T Δ
• all: {L : LO.FirstOrder.Language} → [inst : (k : ℕ) → DecidableEq (L.Func k)] → [inst_1 : (k : ℕ) → DecidableEq (L.Rel k)] → {T : LO.FirstOrder.Theory L} → {Δ : Finset (LO.FirstOrder.SyntacticFormula L)} → {φ : LO.FirstOrder.SyntacticSemiformula L 1} → ∀' φ ∈ Δ → LO.FirstOrder.Derivation2 T (insert (LO.FirstOrder.Rewriting.free φ) (Finset.image LO.FirstOrder.Rewriting.shift Δ)) → LO.FirstOrder.Derivation2 T Δ
• ex: {L : LO.FirstOrder.Language} → [inst : (k : ℕ) → DecidableEq (L.Func k)] → [inst_1 : (k : ℕ) → DecidableEq (L.Rel k)] → {T : LO.FirstOrder.Theory L} → {Δ : Finset (LO.FirstOrder.SyntacticFormula L)} → {φ : LO.FirstOrder.SyntacticSemiformula L 1} → ∃' φ ∈ Δ → (t : LO.FirstOrder.SyntacticTerm L) → LO.FirstOrder.Derivation2 T (insert (φ/[t]) Δ) → LO.FirstOrder.Derivation2 T Δ
• wk: {L : LO.FirstOrder.Language} → [inst : (k : ℕ) → DecidableEq (L.Func k)] → [inst_1 : (k : ℕ) → DecidableEq (L.Rel k)] → {T : LO.FirstOrder.Theory L} → {Δ Γ : Finset (LO.FirstOrder.SyntacticFormula L)} → LO.FirstOrder.Derivation2 T Δ → Δ ⊆ Γ → LO.FirstOrder.Derivation2 T Γ
• shift: {L : LO.FirstOrder.Language} → [inst : (k : ℕ) → DecidableEq (L.Func k)] → [inst_1 : (k : ℕ) → DecidableEq (L.Rel k)] → {T : LO.FirstOrder.Theory L} → {Δ : Finset (LO.FirstOrder.SyntacticFormula L)} → LO.FirstOrder.Derivation2 T Δ → LO.FirstOrder.Derivation2 T (Finset.image LO.FirstOrder.Rewriting.shift Δ)
• cut: {L : LO.FirstOrder.Language} → [inst : (k : ℕ) → DecidableEq (L.Func k)] → [inst_1 : (k : ℕ) → DecidableEq (L.Rel k)] → {T : LO.FirstOrder.Theory L} → {Δ : Finset (LO.FirstOrder.SyntacticFormula L)} → {φ : LO.FirstOrder.SyntacticFormula L} → LO.FirstOrder.Derivation2 T (insert φ Δ) → LO.FirstOrder.Derivation2 T (insert (∼φ) Δ) → LO.FirstOrder.Derivation2 T Δ

def LO.FirstOrder.«term_⊢₂_» : Lean.TrailingParserDescr

Equations

• LO.FirstOrder.«term_⊢₂_» = Lean.ParserDescr.trailingNode `LO.FirstOrder.«term_⊢₂_» 45 46 (Lean.ParserDescr.binary `andthen (Lean.ParserDescr.symbol " ⊢₂ ") (Lean.ParserDescr.cat `term 46))

@[reducible, inline]

abbrev LO.FirstOrder.Derivable2 {L : LO.FirstOrder.Language} [(k : ℕ) → DecidableEq (L.Func k)] [(k : ℕ) → DecidableEq (L.Rel k)] (T : LO.FirstOrder.Theory L) (Γ : Finset (LO.FirstOrder.SyntacticFormula L)) : Prop

Equations

• LO.FirstOrder.Derivable2 T Γ = Nonempty (LO.FirstOrder.Derivation2 T Γ)

def LO.FirstOrder.«term_⊢₂!_» : Lean.TrailingParserDescr

Equations

• LO.FirstOrder.«term_⊢₂!_» = Lean.ParserDescr.trailingNode `LO.FirstOrder.«term_⊢₂!_» 45 46 (Lean.ParserDescr.binary `andthen (Lean.ParserDescr.symbol " ⊢₂! ") (Lean.ParserDescr.cat `term 46))

@[reducible, inline]

abbrev LO.FirstOrder.Derivable2SingleConseq {L : LO.FirstOrder.Language} [(k : ℕ) → DecidableEq (L.Func k)] [(k : ℕ) → DecidableEq (L.Rel k)] (T : LO.FirstOrder.Theory L) (φ : LO.FirstOrder.SyntacticFormula L) : Prop

Equations

• LO.FirstOrder.Derivable2SingleConseq T φ = LO.FirstOrder.Derivable2 T {φ}

def LO.FirstOrder.«term_⊢₂.!_» : Lean.TrailingParserDescr

Equations

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

theorem LO.FirstOrder.shifts_toFinset_eq_image_shift {L : LO.FirstOrder.Language} [(k : ℕ) → DecidableEq (L.Func k)] [(k : ℕ) → DecidableEq (L.Rel k)] (Δ : LO.FirstOrder.Sequent L) : (LO.FirstOrder.Rewriting.shifts Δ).toFinset = Finset.image LO.FirstOrder.Rewriting.shift (List.toFinset Δ)

def LO.FirstOrder.Derivation.toDerivation2 {L : LO.FirstOrder.Language} [(k : ℕ) → DecidableEq (L.Func k)] [(k : ℕ) → DecidableEq (L.Rel k)] (T : LO.FirstOrder.Theory L) {Γ : LO.FirstOrder.Sequent L} : T ⟹ Γ → LO.FirstOrder.Derivation2 T (List.toFinset Γ)

Equations

• One or more equations did not get rendered due to their size.
• LO.FirstOrder.Derivation.toDerivation2 T (LO.FirstOrder.Derivation.root h) = LO.FirstOrder.Derivation2.root φ h ⋯
• LO.FirstOrder.Derivation.toDerivation2 T (LO.FirstOrder.Derivation.verum Δ) = LO.FirstOrder.Derivation2.verum ⋯
• LO.FirstOrder.Derivation.toDerivation2 T (dp.and dq) = LO.FirstOrder.Derivation2.and ⋯ ((LO.FirstOrder.Derivation.toDerivation2 T dp).wk ⋯) ((LO.FirstOrder.Derivation.toDerivation2 T dq).wk ⋯)
• LO.FirstOrder.Derivation.toDerivation2 T dpq.or = LO.FirstOrder.Derivation2.or ⋯ ((LO.FirstOrder.Derivation.toDerivation2 T dpq).wk ⋯)
• LO.FirstOrder.Derivation.toDerivation2 T dp.all = LO.FirstOrder.Derivation2.all ⋯ ((LO.FirstOrder.Derivation.toDerivation2 T dp).wk ⋯)
• LO.FirstOrder.Derivation.toDerivation2 T (LO.FirstOrder.Derivation.ex t dp) = LO.FirstOrder.Derivation2.ex ⋯ t ((LO.FirstOrder.Derivation.toDerivation2 T dp).wk ⋯)
• LO.FirstOrder.Derivation.toDerivation2 T (d.wk h) = (LO.FirstOrder.Derivation.toDerivation2 T d).wk ⋯
• LO.FirstOrder.Derivation.toDerivation2 T (d₁.cut d₂) = ((LO.FirstOrder.Derivation.toDerivation2 T d₁).wk ⋯).cut ((LO.FirstOrder.Derivation.toDerivation2 T d₂).wk ⋯)

noncomputable def LO.FirstOrder.Derivation2.toDerivation {L : LO.FirstOrder.Language} [(k : ℕ) → DecidableEq (L.Func k)] [(k : ℕ) → DecidableEq (L.Rel k)] {T : LO.FirstOrder.Theory L} {Γ : Finset (LO.FirstOrder.SyntacticFormula L)} : LO.FirstOrder.Derivation2 T Γ → T ⟹ Γ.toList

theorem LO.FirstOrder.derivable_iff_derivable2 {L : LO.FirstOrder.Language} [(k : ℕ) → DecidableEq (L.Func k)] [(k : ℕ) → DecidableEq (L.Rel k)] {T : LO.FirstOrder.Theory L} {Γ : List (LO.FirstOrder.SyntacticFormula L)} : T ⟹! Γ ↔ LO.FirstOrder.Derivable2 T Γ.toFinset

def LO.FirstOrder.provable_iff_derivable2 {L : LO.FirstOrder.Language} [(k : ℕ) → DecidableEq (L.Func k)] [(k : ℕ) → DecidableEq (L.Rel k)] {T : LO.FirstOrder.Theory L} {φ : LO.FirstOrder.SyntacticFormula L} : T ⊢! φ ↔ LO.FirstOrder.Derivable2SingleConseq T φ

Equations

• ⋯ = ⋯