theorem LO.Modal.Standard.axiomTc_GTranslate! {α : Type u} [DecidableEq α] {mΛ : LO.Modal.Standard.DeductionParameter α} {p : LO.Propositional.Superintuitionistic.Formula α} [LO.System.K4 mΛ] : mΛ ⊢! pᵍ ⟶ □pᵍ

instance LO.Modal.Standard.instK4DeductionParameterFormulaOfS4 {α : Type u} {mΛ : LO.Modal.Standard.DeductionParameter α} [LO.System.S4 mΛ] : LO.System.K4 mΛ

Equations

• LO.Modal.Standard.instK4DeductionParameterFormulaOfS4 = LO.System.K4.mk

theorem LO.Modal.Standard.provable_efq_of_provable_S4 {α : Type u} [DecidableEq α] {p : LO.Propositional.Superintuitionistic.Formula α} (h : 𝐈𝐧𝐭 ⊢! p) : 𝐒𝟒 ⊢! pᵍ

theorem LO.Modal.Standard.provable_S4_of_provable_efq {α : Type u} [DecidableEq α] [Inhabited α] [Encodable α] {p : LO.Propositional.Superintuitionistic.Formula α} : 𝐒𝟒 ⊢! pᵍ → 𝐈𝐧𝐭 ⊢! p

theorem LO.Modal.Standard.provable_efq_iff_provable_S4 {α : Type u} [DecidableEq α] [Inhabited α] [Encodable α] {p : LO.Propositional.Superintuitionistic.Formula α} : 𝐈𝐧𝐭 ⊢! p ↔ 𝐒𝟒 ⊢! pᵍ

a.k.a. Gödel-McKinsey-Tarski Theorem

instance LO.Modal.Standard.instModalCompanionIntuitionisticS4 {α : Type u} [DecidableEq α] [Inhabited α] [Encodable α] : LO.Modal.Standard.ModalCompanion 𝐈𝐧𝐭 𝐒𝟒

Equations

• LO.Modal.Standard.instModalCompanionIntuitionisticS4 = { companion := ⋯ }

theorem LO.Modal.Standard.dp_of_mdp {α : Type u} [DecidableEq α] {iΛ : LO.Propositional.Superintuitionistic.DeductionParameter α} {mΛ : LO.Modal.Standard.DeductionParameter α} {p : LO.Propositional.Superintuitionistic.Formula α} {q : LO.Propositional.Superintuitionistic.Formula α} [LO.System.ModalDisjunctive mΛ] [LO.Modal.Standard.ModalCompanion iΛ mΛ] [LO.System.S4 mΛ] : iΛ ⊢! p ⋎ q → iΛ ⊢! p ∨ iΛ ⊢! q

theorem LO.Modal.Standard.disjunctive_of_modalDisjunctive {α : Type u} [DecidableEq α] {iΛ : LO.Propositional.Superintuitionistic.DeductionParameter α} {mΛ : LO.Modal.Standard.DeductionParameter α} [LO.System.ModalDisjunctive mΛ] [LO.Modal.Standard.ModalCompanion iΛ mΛ] [LO.System.S4 mΛ] : LO.System.Disjunctive iΛ