-
Notifications
You must be signed in to change notification settings - Fork 0
Commit
This commit does not belong to any branch on this repository, and may belong to a fork outside of the repository.
- Loading branch information
Showing
2 changed files
with
132 additions
and
51 deletions.
There are no files selected for viewing
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,71 @@ | ||
| import LabelledSystem.Gentzen.Basic | ||
|
|
||
| namespace LO.Modal.Labelled.Gentzen | ||
|
|
||
| -- def Sequent.ofFormula (φ : Formula ℕ) : Sequent := ⟨∅, ∅⟩ ⟹ ⟨{0 ∶ φ}, ∅⟩ | ||
|
|
||
| variable {x : Label} {φ ψ χ : Formula ℕ} | ||
|
|
||
| def imply₁ : ⊢ᵍ ⟨∅, ∅⟩ ⟹ ⟨{x ∶ φ ➝ ψ ➝ φ}, ∅⟩ := by | ||
| apply impR (Δ := ⟨_, _⟩); | ||
| apply impR (Δ := ⟨_, _⟩); | ||
| have e : (x ∶ ψ) ::ₘ {x ∶ φ} = (x ∶ φ) ::ₘ {x ∶ ψ} := by sorry; | ||
| exact initFml' x φ (by simp) (by simp); | ||
|
|
||
| def imply₂ : ⊢ᵍ ⟨∅, ∅⟩ ⟹ ⟨{x ∶ (φ ➝ ψ ➝ χ) ➝ (φ ➝ ψ) ➝ φ ➝ χ}, ∅⟩ := by | ||
| apply impR (Δ := ⟨_, _⟩); | ||
| apply impR (Δ := ⟨_, _⟩); | ||
| apply impR (Δ := ⟨_, _⟩); | ||
| rw [Multiset.cons_swap]; | ||
| apply impL (Γ := ⟨{x ∶ φ, x ∶ φ ➝ ψ ➝ χ}, _⟩); | ||
| . exact initFml' x φ (by simp) (by simp); | ||
| . suffices ⊢ᵍ ⟨(x ∶ φ ➝ ψ ➝ χ) ::ₘ {x ∶ ψ, x ∶ φ}, ∅⟩ ⟹ ⟨{x ∶ χ}, ∅⟩ by | ||
| have e : (x ∶ ψ) ::ₘ (x ∶ φ) ::ₘ {x ∶ φ ➝ ψ ➝ χ} = (x ∶ φ ➝ ψ ➝ χ) ::ₘ {x ∶ ψ, x ∶ φ} := by sorry; | ||
| simpa [e]; | ||
| apply impL (Γ := ⟨_, _⟩); | ||
| . exact initFml' x φ (by simp) (by simp); | ||
| . apply impL (Γ := ⟨_, _⟩); | ||
| . exact initFml' x ψ (by simp) (by simp); | ||
| . exact initFml' x χ (by simp) (by simp); | ||
|
|
||
| def elimContra : ⊢ᵍ ⟨∅, ∅⟩ ⟹ ⟨{x ∶ (∼ψ ➝ ∼φ) ➝ (φ ➝ ψ)}, ∅⟩ := by | ||
| apply impR (Δ := ⟨_, _⟩); | ||
| apply impR (Δ := ⟨_, _⟩); | ||
| rw [Multiset.cons_swap]; | ||
| apply impL (Γ := ⟨{x ∶ φ}, _⟩); | ||
| . -- TODO: `⊢ᵍ Γ ⟹ ⟨(x ∶ ∼ψ) ::ₘ {x ∶ ψ}, Δ.rels⟩` | ||
| apply impR (Δ := ⟨_, _⟩); | ||
| exact initFml' x ψ (by simp) (by simp); | ||
| . -- TODO: `⊢ᵍ ⟨(x ∶ ∼φ) ::ₘ {x ∶ φ} :: Γ.fmls, ∅⟩ ⟹ Δ` | ||
| apply impL (Γ := ⟨{x ∶ φ}, _⟩); | ||
| . exact initFml' x φ (by simp) (by simp); | ||
| . exact initBot; | ||
|
|
||
| def axiomK : ⊢ᵍ ⟨∅, ∅⟩ ⟹ ⟨{x ∶ □(φ ➝ ψ) ➝ □φ ➝ □ψ}, ∅⟩ := by | ||
| letI y : Label := x + 1; | ||
| apply impR (Δ := ⟨_, _⟩); | ||
| apply impR; | ||
| apply boxR (y := y) (by simp [y]) (by simp) (by simp); | ||
| suffices ⊢ᵍ (⟨(x ∶ □φ) ::ₘ {x ∶ □(φ ➝ ψ)}, {(x, y)}⟩ ⟹ ⟨{y ∶ ψ}, ∅⟩) by simpa; | ||
| apply boxL (Γ := ⟨_, _⟩); | ||
| suffices ⊢ᵍ (⟨(x ∶ □(φ ➝ ψ)) ::ₘ (y ∶ φ) ::ₘ {(x ∶ □φ)}, {(x, y)}⟩ ⟹ ⟨{y ∶ ψ}, ∅⟩) by | ||
| have e : (x ∶ □(φ ➝ ψ)) ::ₘ (y ∶ φ) ::ₘ {x ∶ □φ} = (x ∶ □φ) ::ₘ (y ∶ φ) ::ₘ {x ∶ □(φ ➝ ψ)} := by sorry; | ||
| simpa [e]; | ||
| apply boxL (Γ := ⟨{y ∶ φ, x ∶ □φ}, _⟩); | ||
| suffices ⊢ᵍ (⟨(y ∶ φ ➝ ψ) ::ₘ {y ∶ φ, x ∶ □φ, x ∶ □(φ ➝ ψ)}, {(x, y)}⟩ ⟹ ⟨{y ∶ ψ}, ∅⟩) by | ||
| have e : (x ∶ □(φ ➝ ψ)) ::ₘ (y ∶ φ ➝ ψ) ::ₘ (y ∶ φ) ::ₘ {x ∶ □φ} = (y ∶ φ ➝ ψ) ::ₘ {y ∶ φ, x ∶ □φ, x ∶ □(φ ➝ ψ)} := by sorry; | ||
| simpa [e]; | ||
| apply impL (Γ := ⟨_, _⟩); | ||
| . simpa using initFml (Γ := ⟨_, _⟩) (Δ := ⟨_, _⟩); | ||
| . simpa using initFml (Γ := ⟨_, _⟩) (Δ := ⟨_, _⟩); | ||
|
|
||
| def mdp (d₁ : ⊢ᵍ ⟨∅, ∅⟩ ⟹ ⟨{x ∶ φ ➝ ψ}, ∅⟩) (d₂ : ⊢ᵍ ⟨∅, ∅⟩ ⟹ ⟨{x ∶ φ}, ∅⟩) : ⊢ᵍ ⟨∅, ∅⟩ ⟹ ⟨{x ∶ ψ}, ∅⟩ := by | ||
| simpa using cutFml (Δ₁ := ⟨∅, ∅⟩) (Δ₂ := ⟨{x ∶ ψ}, ∅⟩) d₂ $ implyRInv (Δ := ⟨∅, ∅⟩) d₁; | ||
|
|
||
| def necessitation (d : ⊢ᵍ ⟨∅, ∅⟩ ⟹ ⟨{x ∶ φ}, ∅⟩) : ⊢ᵍ ⟨∅, ∅⟩ ⟹ ⟨{x ∶ □φ}, ∅⟩ := by | ||
| letI y : Label := x + 1; | ||
| apply boxR (Δ := ⟨∅, ∅⟩) (y := y) (by simp [y]) (by simp) (by simp); | ||
| apply wkRelL; | ||
| simpa [SequentPart.replaceLabel, LabelledFormula.labelReplace, LabelReplace.specific] using replaceLabel d (x ⧸ y); | ||
|
|
||
| end LO.Modal.Labelled.Gentzen |