@[reducible, inline]
Equations
- LO.Modal.Theory.Consistent Λ T = T *⊬[Λ] ⊥
Instances For
@[reducible, inline]
abbrev
LO.Modal.Theory.Inconsistent
{α : Type u_1}
(Λ : LO.Modal.Hilbert α)
(T : LO.Modal.Theory α)
:
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)), (∀ q ∈ Γ, 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)), (∀ q ∈ Γ, q ∈ T) ∧ Γ ⊢[Λ]! ⊥
@[simp]
theorem
LO.Modal.Theory.self_consistent
{α : Type u_1}
[DecidableEq α]
{Λ : LO.Modal.Hilbert α}
[Λ_consis : LO.System.Consistent Λ]
:
theorem
LO.Modal.Theory.emptyset_consistent
{α : Type u_1}
[DecidableEq α]
{Λ : LO.Modal.Hilbert α}
[Λ_consis : LO.System.Consistent Λ]
:
theorem
LO.Modal.Theory.union_consistent
{α : Type u_1}
{Λ : LO.Modal.Hilbert α}
{T₁ : LO.Modal.Theory α}
{T₂ : LO.Modal.Theory α}
:
LO.Modal.Theory.Consistent Λ (T₁ ∪ T₂) → LO.Modal.Theory.Consistent Λ T₁ ∧ LO.Modal.Theory.Consistent Λ T₂
theorem
LO.Modal.Theory.union_not_consistent
{α : Type u_1}
{Λ : LO.Modal.Hilbert α}
{T₁ : LO.Modal.Theory α}
{T₂ : LO.Modal.Theory α}
:
¬LO.Modal.Theory.Consistent Λ T₁ ∨ ¬LO.Modal.Theory.Consistent Λ T₂ → ¬LO.Modal.Theory.Consistent Λ (T₁ ∪ T₂)
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}
:
theorem
LO.Modal.Theory.iff_insert_inconsistent
{α : Type u_1}
[DecidableEq α]
{Λ : LO.Modal.Hilbert α}
{T : LO.Modal.Theory α}
{p : LO.Modal.Formula α}
:
theorem
LO.Modal.Theory.provable_iff_insert_neg_not_consistent
{α : Type u_1}
[DecidableEq α]
{Λ : LO.Modal.Hilbert α}
{T : LO.Modal.Theory α}
{p : LO.Modal.Formula α}
:
theorem
LO.Modal.Theory.unprovable_iff_insert_neg_consistent
{α : Type u_1}
[DecidableEq α]
{Λ : LO.Modal.Hilbert α}
{T : LO.Modal.Theory α}
{p : LO.Modal.Formula α}
:
theorem
LO.Modal.Theory.unprovable_iff_singleton_neg_consistent
{α : Type u_1}
[DecidableEq α]
{Λ : LO.Modal.Hilbert α}
{p : LO.Modal.Formula α}
:
Λ ⊬ p ↔ LO.Modal.Theory.Consistent Λ {∼p}
theorem
LO.Modal.Theory.neg_provable_iff_insert_not_consistent
{α : Type u_1}
[DecidableEq α]
{Λ : LO.Modal.Hilbert α}
{T : LO.Modal.Theory α}
{p : LO.Modal.Formula α}
:
theorem
LO.Modal.Theory.neg_unprovable_iff_insert_consistent
{α : Type u_1}
[DecidableEq α]
{Λ : LO.Modal.Hilbert α}
{T : LO.Modal.Theory α}
{p : LO.Modal.Formula α}
:
theorem
LO.Modal.Theory.unprovable_iff_singleton_consistent
{α : Type u_1}
[DecidableEq α]
{Λ : LO.Modal.Hilbert α}
{p : LO.Modal.Formula α}
:
Λ ⊬ ∼p ↔ LO.Modal.Theory.Consistent Λ {p}
theorem
LO.Modal.Theory.unprovable_falsum
{α : Type u_1}
{Λ : LO.Modal.Hilbert α}
{T : LO.Modal.Theory α}
(T_consis : LO.Modal.Theory.Consistent Λ T)
:
theorem
LO.Modal.Theory.unprovable_either
{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}
:
theorem
LO.Modal.Theory.not_mem_falsum_of_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)
:
⊥ ∉ T
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.either_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)
:
LO.Modal.Theory.Consistent Λ (insert p T) ∨ LO.Modal.Theory.Consistent Λ (insert (∼p) T)
theorem
LO.Modal.Theory.exists_maximal_consistent_theory
{α : Type u_1}
[DecidableEq α]
{Λ : LO.Modal.Hilbert α}
{T : LO.Modal.Theory α}
(T_consis : LO.Modal.Theory.Consistent Λ T)
:
∃ (Z : LO.Modal.Theory α),
LO.Modal.Theory.Consistent Λ Z ∧ T ⊆ Z ∧ ∀ (U : LO.Modal.Theory α), LO.Modal.Theory.Consistent Λ U → Z ⊆ U → U = Z
theorem
LO.Modal.Theory.lindenbaum
{α : Type u_1}
[DecidableEq α]
{Λ : LO.Modal.Hilbert α}
{T : LO.Modal.Theory α}
(T_consis : LO.Modal.Theory.Consistent Λ T)
:
∃ (Z : LO.Modal.Theory α),
LO.Modal.Theory.Consistent Λ Z ∧ T ⊆ Z ∧ ∀ (U : LO.Modal.Theory α), LO.Modal.Theory.Consistent Λ U → Z ⊆ U → U = Z
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)}, ((∀ p ∈ Γ₁, p ∈ T₁) ∧ ∀ p ∈ Γ₂, p ∈ T₂) → Λ ⊬ ⋀Γ₁ ⋏ ⋀Γ₂ ➝ ⊥)
:
LO.Modal.Theory.Consistent Λ (T₁ ∪ T₂)
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)},
((∀ p ∈ Γ₁, p ∈ T₁) ∧ (∀ p ∈ Γ₂, p ∈ T₂) ∧ ∀ p ∈ Γ₃, 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)
:
p ∉ T
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)
:
∼p ∉ T
- theory : LO.Modal.Theory α
- consistent : LO.Modal.Theory.Consistent Λ self.theory
- maximal : ∀ {U : LO.Modal.Theory α}, self.theory ⊂ U → ¬LO.Modal.Theory.Consistent Λ U
Instances For
theorem
LO.Modal.MaximalConsistentTheory.consistent
{α : Type u_1}
{Λ : LO.Modal.Hilbert α}
(self : LO.Modal.MaximalConsistentTheory Λ)
:
LO.Modal.Theory.Consistent Λ self.theory
theorem
LO.Modal.MaximalConsistentTheory.maximal
{α : Type u_1}
{Λ : LO.Modal.Hilbert α}
(self : LO.Modal.MaximalConsistentTheory Λ)
{U : LO.Modal.Theory α}
:
self.theory ⊂ U → ¬LO.Modal.Theory.Consistent Λ U
Alias of LO.Modal.MaximalConsistentTheory
.
Instances For
theorem
LO.Modal.MaximalConsistentTheory.exists_maximal_Lconsistented_theory
{α : Type u_1}
[DecidableEq α]
{Λ : LO.Modal.Hilbert α}
{T : LO.Modal.Theory α}
(consisT : LO.Modal.Theory.Consistent Λ T)
:
∃ (Ω : LO.Modal.MCT Λ), T ⊆ Ω.theory
theorem
LO.Modal.MaximalConsistentTheory.lindenbaum
{α : Type u_1}
[DecidableEq α]
{Λ : LO.Modal.Hilbert α}
{T : LO.Modal.Theory α}
(consisT : LO.Modal.Theory.Consistent Λ T)
:
∃ (Ω : LO.Modal.MCT Λ), T ⊆ Ω.theory
Alias of LO.Modal.MaximalConsistentTheory.exists_maximal_Lconsistented_theory
.
instance
LO.Modal.MaximalConsistentTheory.instNonemptyMCTOfConsistentFormulaHilbert
{α : Type u_1}
[DecidableEq α]
{Λ : LO.Modal.Hilbert α}
[LO.System.Consistent Λ]
:
Nonempty (LO.Modal.MCT Λ)
Equations
- ⋯ = ⋯
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)
:
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 : p ∉ Ω.theory)
:
¬LO.Modal.Theory.Consistent Λ (insert p Ω.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}
:
theorem
LO.Modal.MaximalConsistentTheory.subset_axiomset
{α : Type u_1}
[DecidableEq α]
{Λ : LO.Modal.Hilbert α}
{Ω : LO.Modal.MCT Λ}
:
@[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}
:
@[simp]
theorem
LO.Modal.MaximalConsistentTheory.unprovable_falsum
{α : Type u_1}
[DecidableEq α]
{Λ : LO.Modal.Hilbert α}
{Ω : LO.Modal.MCT Λ}
:
@[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}
:
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}
:
@[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}
:
@[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}
:
@[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}
:
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}
:
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}
:
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}
:
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 : ∀ p ∈ Ω₁.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.theory → p ∈ Ω₁.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)
:
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}
:
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]
:
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}
:
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]
:
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}
:
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]
:
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}
:
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]
:
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}
:
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]
:
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 : ℕ}
:
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)}
:
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}
:
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)}
: