Theorem: LO.Modal.Standard.provable_S4_of_provable_efq
Proof state before:
case intro.intro.intro.intro.intro.left
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
inst✝ : Encodable α
i𝓓 : Superintuitionistic.DeductionParameter α
m𝓓 : DeductionParameter α
p q r : Superintuitionistic.Formula α
h✝ : ¬𝐈𝐧𝐭 ⊢! p
iF : Superintuitionistic.Kripke.Frame' α
left✝ : iF ∈ 𝔽(𝗘𝗙𝗤)
iV : Superintuitionistic.Kripke.Valuation iF.World α
iV_hereditary : ∀ {w₁ w₂ : iF.World}, w₁ ≺ w₂ → ∀ {a : α}, iV w₁ a → iV w₂ a
w : { Frame := iF, Valuation := iV, hereditary := ⋯ }.World
h : ¬Satisfies { Frame := iF, Valuation := iV, hereditary := ⋯ } w p
M : Kripke.Model α := { Frame := { World := iF.World, World_nonempty := ⋯, Rel := iF.Rel }, Valuation := iV }
h₁ :
  ∀ (q : Superintuitionistic.Formula α) (v : { Frame := iF, Valuation := iV, hereditary := ⋯ }.World),
    Satisfies { Frame := iF, Valuation := iV, hereditary := ⋯ } v q ↔ Formula.Kripke.Satisfies M v (qᵍ)
this : ¬Formula.Kripke.Satisfies M w (pᵍ)
⊢ M.Frame ∈ 𝔽(𝗞 ∪ 𝗧 ∪ 𝟰)

case intro.intro.intro.intro.intro.right
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
inst✝ : Encodable α
i𝓓 : Superintuitionistic.DeductionParameter α
m𝓓 : DeductionParameter α
p q r : Superintuitionistic.Formula α
h✝ : ¬𝐈𝐧𝐭 ⊢! p
iF : Superintuitionistic.Kripke.Frame' α
left✝ : iF ∈ 𝔽(𝗘𝗙𝗤)
iV : Superintuitionistic.Kripke.Valuation iF.World α
iV_hereditary : ∀ {w₁ w₂ : iF.World}, w₁ ≺ w₂ → ∀ {a : α}, iV w₁ a → iV w₂ a
w : { Frame := iF, Valuation := iV, hereditary := ⋯ }.World
h : ¬Satisfies { Frame := iF, Valuation := iV, hereditary := ⋯ } w p
M : Kripke.Model α := { Frame := { World := iF.World, World_nonempty := ⋯, Rel := iF.Rel }, Valuation := iV }
h₁ :
  ∀ (q : Superintuitionistic.Formula α) (v : { Frame := iF, Valuation := iV, hereditary := ⋯ }.World),
    Satisfies { Frame := iF, Valuation := iV, hereditary := ⋯ } v q ↔ Formula.Kripke.Satisfies M v (qᵍ)
this : ¬Formula.Kripke.Satisfies M w (pᵍ)
⊢ ∃ x x_1, ¬Formula.Kripke.Satisfies { Frame := M.Frame, Valuation := x } x_1 (pᵍ)
Tactic:
. apply Modal.Standard.Kripke.iff_definability_memAxiomSetFrameClass
    (show Kripke.Definability _ (λ F => Reflexive F.Rel ∧ Transitive F.Rel) by simpa using instGeachDefinability (L := 𝐒𝟒))
    |>.mpr;
  constructor;
  . exact iF.Rel_refl;
  . exact iF.Rel_trans;
Proof state after:
case intro.intro.intro.intro.intro.right
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
inst✝ : Encodable α
i𝓓 : Superintuitionistic.DeductionParameter α
m𝓓 : DeductionParameter α
p q r : Superintuitionistic.Formula α
h✝ : ¬𝐈𝐧𝐭 ⊢! p
iF : Superintuitionistic.Kripke.Frame' α
left✝ : iF ∈ 𝔽(𝗘𝗙𝗤)
iV : Superintuitionistic.Kripke.Valuation iF.World α
iV_hereditary : ∀ {w₁ w₂ : iF.World}, w₁ ≺ w₂ → ∀ {a : α}, iV w₁ a → iV w₂ a
w : { Frame := iF, Valuation := iV, hereditary := ⋯ }.World
h : ¬Satisfies { Frame := iF, Valuation := iV, hereditary := ⋯ } w p
M : Kripke.Model α := { Frame := { World := iF.World, World_nonempty := ⋯, Rel := iF.Rel }, Valuation := iV }
h₁ :
  ∀ (q : Superintuitionistic.Formula α) (v : { Frame := iF, Valuation := iV, hereditary := ⋯ }.World),
    Satisfies { Frame := iF, Valuation := iV, hereditary := ⋯ } v q ↔ Formula.Kripke.Satisfies M v (qᵍ)
this : ¬Formula.Kripke.Satisfies M w (pᵍ)
⊢ ∃ x x_1, ¬Formula.Kripke.Satisfies { Frame := M.Frame, Valuation := x } x_1 (pᵍ)