Deduction of PTL Axioms from the FL Axiom System

⊢	(f ⊃ g) ⊃ f ⊃ g	A1
⊢	f ⊃ f	A2
⊢	(f ⊃ g) ⊃ f ⊃ g	A3
⊢	f ⊃ f ∧ f	A4
⊢	(f ⊃ f) ⊃ f ⊃ f	A5
	f is a tautology ⇒ ⊢ f	R1
⊢	f ⊃ g, ⊢ f ⇒ ⊢ g	R2
⊢	f ⇒ ⊢ f	R3

Table 2: Modiﬁed version of Pnueli’s complete axiom system

⊢	All FL tautologies	FLTaut
⊢	X ≡⟨skip⟩f	FL2
⊢	f ≡⟨skip^∗⟩f	FL3
⊢	⟨w?⟩f ≡ w ∧ f	FL4
⊢	⟨E₀ ∨ E₁⟩f ≡⟨E₀⟩f ∨⟨E₁⟩f	FL5
⊢	⟨E₀ ; E₁⟩f ≡⟨E₀⟩⟨E₁⟩f	FL6
⊢	⟨E⟩(f ∨ g) ⊃⟨E⟩f ∨⟨E⟩g	FL7
⊢	⟨E^∗⟩f ≡ f ∨⟨E ; E^∗⟩f	FL8
⊢	(f ⊃ g) ⊃⟨E⟩f ⊃⟨E⟩g	FL9
⊢	f ⊃ f	FL10
⊢	(f ⊃ f) ∧ f ⊃ f	FL11
⊢	empty	FL12
⊢	f ⊃ g, ⊢ f ⇒ ⊢ g	FLMP
⊢	f ⇒ ⊢ f	FLBoxGen
⊢	⟨E₀⟩empty ⊃⟨E₀⟩empty ⇒ ⊢ ⟨E₀⟩f ⊃⟨E₁⟩f	FInf3
⊢	(more ∧⟨E₀⟩empty)⟨E₁⟩empty ⇒ ⊢ ⟨E₀^∗⟩f ⊃⟨E₁^∗⟩f	FInf4

Table 3: Axiom system for FL

This appendix contains various FL theorems and their deductions. These include ones corresponding to some of the PTL axioms in Table 2. Most of the PTL axioms and inference rules have identical or nearly identical versions in the FL axiom system in Table 3. The three exceptions are Axioms A1, A3 and A4. We will look at each of them in turn as FL theorems FBoxImpDist, FNextImpDist and FBoxImpNowAndWeakNext, respectively. The trickiest is Axiom A1. The symbol ⊢ as used here always refers to ⊢ _FL. None of the FE formulas occurring in the proofs contain variables and therefore the proofs also ensure well-formed FLV theorems and derived inference rules for any V.

5 Deduction of PTL Axioms from the FL Axiom System