Documentation

Foundation.Modal.ConsistentTheory

@[reducible, inline]
Equations
Instances For
    @[reducible, inline]
    Equations
    Instances For
      theorem LO.Modal.Theory.def_consistent {α : Type u_1} {Λ : LO.Modal.Hilbert α} {T : LO.Modal.Theory α} :
      LO.Modal.Theory.Consistent Λ T ∀ (Γ : List (LO.Modal.Formula α)), (qΓ, q T)Γ ⊬[Λ]
      theorem LO.Modal.Theory.def_inconsistent {α : Type u_1} {Λ : LO.Modal.Hilbert α} {T : LO.Modal.Theory α} :
      ¬LO.Modal.Theory.Consistent Λ T ∃ (Γ : List (LO.Modal.Formula α)), (qΓ, q T) Γ ⊢[Λ]!
      theorem LO.Modal.Theory.iff_insert_consistent {α : Type u_1} [DecidableEq α] {Λ : LO.Modal.Hilbert α} {T : LO.Modal.Theory α} {p : LO.Modal.Formula α} :
      LO.Modal.Theory.Consistent Λ (insert p T) ∀ {Γ : List (LO.Modal.Formula α)}, (qΓ, q T)Λ p Γ
      theorem LO.Modal.Theory.not_singleton_consistent {α : Type u_1} [DecidableEq α] {Λ : LO.Modal.Hilbert α} {T : LO.Modal.Theory α} (T_consis : LO.Modal.Theory.Consistent Λ T) {p : LO.Modal.Formula α} [Λ.HasNecessitation] (h : p T) :
      theorem LO.Modal.Theory.intro_union_consistent {α : Type u_1} [DecidableEq α] {Λ : LO.Modal.Hilbert α} {T₁ : LO.Modal.Theory α} {T₂ : LO.Modal.Theory α} (h : ∀ {Γ₁ Γ₂ : List (LO.Modal.Formula α)}, ((pΓ₁, p T₁) pΓ₂, p T₂)Λ Γ₁ Γ₂ ) :
      theorem LO.Modal.Theory.intro_triunion_consistent {α : Type u_1} [DecidableEq α] {Λ : LO.Modal.Hilbert α} {T₁ : LO.Modal.Theory α} {T₂ : LO.Modal.Theory α} {T₃ : LO.Modal.Theory α} (h : ∀ {Γ₁ Γ₂ Γ₃ : List (LO.Modal.Formula α)}, ((pΓ₁, p T₁) (pΓ₂, p T₂) pΓ₃, p T₃)Λ Γ₁ Γ₂ Γ₃ ) :
      LO.Modal.Theory.Consistent Λ (T₁ T₂ T₃)
      theorem LO.Modal.Theory.not_mem_of_mem_neg {α : Type u_1} [DecidableEq α] {Λ : LO.Modal.Hilbert α} {T : LO.Modal.Theory α} (T_consis : LO.Modal.Theory.Consistent Λ T) {p : LO.Modal.Formula α} (h : p T) :
      pT
      theorem LO.Modal.Theory.not_mem_neg_of_mem {α : Type u_1} [DecidableEq α] {Λ : LO.Modal.Hilbert α} {T : LO.Modal.Theory α} (T_consis : LO.Modal.Theory.Consistent Λ T) {p : LO.Modal.Formula α} (h : p T) :
      pT
      structure LO.Modal.MaximalConsistentTheory {α : Type u_1} (Λ : LO.Modal.Hilbert α) :
      Type u_1
      Instances For
        theorem LO.Modal.MaximalConsistentTheory.either_mem {α : Type u_1} [DecidableEq α] {Λ : LO.Modal.Hilbert α} (Ω : LO.Modal.MCT Λ) (p : LO.Modal.Formula α) :
        p Ω.theory p Ω.theory
        theorem LO.Modal.MaximalConsistentTheory.maximal' {α : Type u_1} {Λ : LO.Modal.Hilbert α} {Ω : LO.Modal.MCT Λ} {p : LO.Modal.Formula α} (hp : pΩ.theory) :
        theorem LO.Modal.MaximalConsistentTheory.membership_iff {α : Type u_1} [DecidableEq α] {Λ : LO.Modal.Hilbert α} {Ω : LO.Modal.MCT Λ} {p : LO.Modal.Formula α} :
        p Ω.theory Ω.theory *⊢[Λ]! p
        @[simp]
        theorem LO.Modal.MaximalConsistentTheory.not_mem_falsum {α : Type u_1} [DecidableEq α] {Λ : LO.Modal.Hilbert α} {Ω : LO.Modal.MCT Λ} :
        Ω.theory
        @[simp]
        theorem LO.Modal.MaximalConsistentTheory.mem_verum {α : Type u_1} [DecidableEq α] {Λ : LO.Modal.Hilbert α} {Ω : LO.Modal.MCT Λ} :
        Ω.theory
        @[simp]
        theorem LO.Modal.MaximalConsistentTheory.iff_mem_neg {α : Type u_1} [DecidableEq α] {Λ : LO.Modal.Hilbert α} {Ω : LO.Modal.MCT Λ} {p : LO.Modal.Formula α} :
        p Ω.theory pΩ.theory
        theorem LO.Modal.MaximalConsistentTheory.iff_mem_negneg {α : Type u_1} [DecidableEq α] {Λ : LO.Modal.Hilbert α} {Ω : LO.Modal.MCT Λ} {p : LO.Modal.Formula α} :
        p Ω.theory p Ω.theory
        @[simp]
        theorem LO.Modal.MaximalConsistentTheory.iff_mem_imp {α : Type u_1} [DecidableEq α] {Λ : LO.Modal.Hilbert α} {Ω : LO.Modal.MCT Λ} {p : LO.Modal.Formula α} {q : LO.Modal.Formula α} :
        p q Ω.theory p Ω.theoryq Ω.theory
        @[simp]
        theorem LO.Modal.MaximalConsistentTheory.iff_mem_and {α : Type u_1} [DecidableEq α] {Λ : LO.Modal.Hilbert α} {Ω : LO.Modal.MCT Λ} {p : LO.Modal.Formula α} {q : LO.Modal.Formula α} :
        p q Ω.theory p Ω.theory q Ω.theory
        @[simp]
        theorem LO.Modal.MaximalConsistentTheory.iff_mem_or {α : Type u_1} [DecidableEq α] {Λ : LO.Modal.Hilbert α} {Ω : LO.Modal.MCT Λ} {p : LO.Modal.Formula α} {q : LO.Modal.Formula α} :
        p q Ω.theory p Ω.theory q Ω.theory
        theorem LO.Modal.MaximalConsistentTheory.iff_congr {α : Type u_1} [DecidableEq α] {Λ : LO.Modal.Hilbert α} {Ω : LO.Modal.MCT Λ} {p : LO.Modal.Formula α} {q : LO.Modal.Formula α} :
        Ω.theory *⊢[Λ]! p q(p Ω.theory q Ω.theory)
        theorem LO.Modal.MaximalConsistentTheory.mem_dn_iff {α : Type u_1} [DecidableEq α] {Λ : LO.Modal.Hilbert α} {Ω : LO.Modal.MCT Λ} {p : LO.Modal.Formula α} :
        p Ω.theory p Ω.theory
        theorem LO.Modal.MaximalConsistentTheory.equality_def {α : Type u_1} {Λ : LO.Modal.Hilbert α} {Ω₁ : LO.Modal.MCT Λ} {Ω₂ : LO.Modal.MCT Λ} :
        Ω₁ = Ω₂ Ω₁.theory = Ω₂.theory
        theorem LO.Modal.MaximalConsistentTheory.intro_equality {α : Type u_1} [DecidableEq α] {Λ : LO.Modal.Hilbert α} {Ω₁ : LO.Modal.MCT Λ} {Ω₂ : LO.Modal.MCT Λ} {h : pΩ₁.theory, p Ω₂.theory} :
        Ω₁ = Ω₂
        theorem LO.Modal.MaximalConsistentTheory.neg_imp {α : Type u_1} [DecidableEq α] {Λ : LO.Modal.Hilbert α} {Ω₁ : LO.Modal.MCT Λ} {Ω₂ : LO.Modal.MCT Λ} {p : LO.Modal.Formula α} {q : LO.Modal.Formula α} (h : q Ω₂.theoryp Ω₁.theory) :
        p Ω₁.theoryq Ω₂.theory
        theorem LO.Modal.MaximalConsistentTheory.neg_iff {α : Type u_1} [DecidableEq α] {Λ : LO.Modal.Hilbert α} {Ω₁ : LO.Modal.MCT Λ} {Ω₂ : LO.Modal.MCT Λ} {p : LO.Modal.Formula α} {q : LO.Modal.Formula α} (h : p Ω₁.theory q Ω₂.theory) :
        p Ω₁.theory q Ω₂.theory
        theorem LO.Modal.MaximalConsistentTheory.iff_mem_multibox {α : Type u_1} [DecidableEq α] {Λ : LO.Modal.Hilbert α} {Ω : LO.Modal.MCT Λ} {p : LO.Modal.Formula α} [Λ.IsNormal] {n : } :
        □^[n]p Ω.theory ∀ {Ω' : LO.Modal.MCT Λ}, □''⁻¹^[n]Ω.theory Ω'.theoryp Ω'.theory
        theorem LO.Modal.MaximalConsistentTheory.iff_mem_box {α : Type u_1} [DecidableEq α] {Λ : LO.Modal.Hilbert α} {Ω : LO.Modal.MCT Λ} {p : LO.Modal.Formula α} [Λ.IsNormal] :
        p Ω.theory ∀ {Ω' : LO.Modal.MCT Λ}, □''⁻¹Ω.theory Ω'.theoryp Ω'.theory
        theorem LO.Modal.MaximalConsistentTheory.multibox_dn_iff {α : Type u_1} [DecidableEq α] {Λ : LO.Modal.Hilbert α} {Ω : LO.Modal.MCT Λ} {p : LO.Modal.Formula α} [Λ.IsNormal] {n : } :
        □^[n](p) Ω.theory □^[n]p Ω.theory
        theorem LO.Modal.MaximalConsistentTheory.box_dn_iff {α : Type u_1} [DecidableEq α] {Λ : LO.Modal.Hilbert α} {Ω : LO.Modal.MCT Λ} {p : LO.Modal.Formula α} [Λ.IsNormal] :
        (p) Ω.theory p Ω.theory
        theorem LO.Modal.MaximalConsistentTheory.mem_multibox_dual {α : Type u_1} [DecidableEq α] {Λ : LO.Modal.Hilbert α} {Ω : LO.Modal.MCT Λ} {p : LO.Modal.Formula α} [Λ.IsNormal] {n : } :
        □^[n]p Ω.theory ◇^[n](p) Ω.theory
        theorem LO.Modal.MaximalConsistentTheory.mem_box_dual {α : Type u_1} [DecidableEq α] {Λ : LO.Modal.Hilbert α} {Ω : LO.Modal.MCT Λ} {p : LO.Modal.Formula α} [Λ.IsNormal] :
        p Ω.theory (p) Ω.theory
        theorem LO.Modal.MaximalConsistentTheory.mem_multidia_dual {α : Type u_1} [DecidableEq α] {Λ : LO.Modal.Hilbert α} {Ω : LO.Modal.MCT Λ} {p : LO.Modal.Formula α} [Λ.IsNormal] {n : } :
        ◇^[n]p Ω.theory □^[n](p) Ω.theory
        theorem LO.Modal.MaximalConsistentTheory.mem_dia_dual {α : Type u_1} [DecidableEq α] {Λ : LO.Modal.Hilbert α} {Ω : LO.Modal.MCT Λ} {p : LO.Modal.Formula α} [Λ.IsNormal] :
        p Ω.theory (p) Ω.theory
        theorem LO.Modal.MaximalConsistentTheory.iff_mem_multidia {α : Type u_1} [DecidableEq α] {Λ : LO.Modal.Hilbert α} {Ω : LO.Modal.MCT Λ} {p : LO.Modal.Formula α} [Λ.IsNormal] {n : } :
        ◇^[n]p Ω.theory ∃ (Ω' : LO.Modal.MCT Λ), □''⁻¹^[n]Ω.theory Ω'.theory p Ω'.theory
        theorem LO.Modal.MaximalConsistentTheory.iff_mem_dia {α : Type u_1} [DecidableEq α] {Λ : LO.Modal.Hilbert α} {Ω : LO.Modal.MCT Λ} {p : LO.Modal.Formula α} [Λ.IsNormal] :
        p Ω.theory ∃ (Ω' : LO.Modal.MCT Λ), □''⁻¹Ω.theory Ω'.theory p Ω'.theory
        theorem LO.Modal.MaximalConsistentTheory.multibox_multidia {α : Type u_1} [DecidableEq α] {Λ : LO.Modal.Hilbert α} {Ω₁ : LO.Modal.MCT Λ} {Ω₂ : LO.Modal.MCT Λ} [Λ.IsNormal] {n : } :
        (∀ {p : LO.Modal.Formula α}, □^[n]p Ω₁.theoryp Ω₂.theory) ∀ {p : LO.Modal.Formula α}, p Ω₂.theory◇^[n]p Ω₁.theory
        theorem LO.Modal.MaximalConsistentTheory.iff_mem_conj {α : Type u_1} [DecidableEq α] {Λ : LO.Modal.Hilbert α} {Ω : LO.Modal.MCT Λ} {Γ : List (LO.Modal.Formula α)} :
        Γ Ω.theory pΓ, p Ω.theory
        theorem LO.Modal.MaximalConsistentTheory.iff_mem_multibox_conj {α : Type u_1} [DecidableEq α] {Λ : LO.Modal.Hilbert α} {Ω : LO.Modal.MCT Λ} [Λ.IsNormal] {Γ : List (LO.Modal.Formula α)} {n : } :
        □^[n]Γ Ω.theory pΓ, □^[n]p Ω.theory
        theorem LO.Modal.MaximalConsistentTheory.iff_mem_box_conj {α : Type u_1} [DecidableEq α] {Λ : LO.Modal.Hilbert α} {Ω : LO.Modal.MCT Λ} [Λ.IsNormal] {Γ : List (LO.Modal.Formula α)} :
        Γ Ω.theory pΓ, p Ω.theory