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5 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 ≡⟨skipf FL2
f ≡⟨skipf FL3
w?f w f FL4
E0 E1f ≡⟨E0f E1f FL5
E0 ; E1f ≡⟨E0⟩⟨E1f FL6
E(f g) Ef Eg FL7
Ef f E ; Ef FL8
(f g) Ef Eg FL9
f f FL10
(f f) f f FL11
empty FL12
f g, f ⇒ ⊢ g FLMP
f ⇒ ⊢ f FLBoxGen
E0empty E0empty ⇒ ⊢ ⟨E0f E1f FInf3
(more E0empty)E1empty ⇒ ⊢ ⟨E0f E1f 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 .

(f g) f g FBoxImpDiamondImpDiamond

Proof:

1
(f g) skipf skipg
2
f ≡⟨skipf
3
g ≡⟨skipg
4
(f g) f g
2, 3,Prop

qed

The following slightly obscure theorem is used in the proof of FBoxImpDist:

(¬g ¬f) f g FBoxContraPosImpDist

Proof:

1
(¬g ¬f) ¬g ¬f
2
(¬g ¬f) ¬¬f ¬¬g
1,Prop
3
(¬g ¬f) f g
2, def. of 

qed

Below is the proof of PTL Axiom A1 as FL theorem FBoxImpDist. In the final step, ImpChain stands for a chain of implications.

(f g) f g FBoxImpDist

Proof:

1
(f g) (¬g ¬f)
2
¬(¬g ¬f) ¬(f g)
1,Prop
3
(¬(¬g ¬f) ¬(f g))
4
(¬(¬g ¬f) ¬(f g))
(f g) (¬g ¬f)
5
(f g) (¬g ¬f)
3, 4,FLMP
6
(¬g ¬f) f g
7
(f g) f g
5, 6,ImpChain

qed

¬f ¬f FNextNotImpNotNext

Proof:

1
f f
2
f ¬¬f
1, def. of 
3
¬f ¬f
2,Prop

qed

Here is a proof of PTL Axiom A3:

(f g) f g FNextImpDist

Proof:

1
skip(¬f g) (skip⟩¬f) (skipg)
2
(¬f g) ≡⟨skip(¬f g)
3
¬f ≡⟨skip⟩¬f
4
g ≡⟨skipg
5
(¬f g) ¬f g
14,Prop
6
¬f ¬f
7
(¬f g) ¬f g
5, 6,Prop
8
(f g) f g
7, def. of 

qed

The remaining proofs are for ultimately deducing PTL Axiom A4 as FL theorem FBoxImpNowAndWeakNext. The following derived rule FRightSkipChopImpSkipChopRule can be readily generalised to allow some arbitrary FE formula in place of skip. In addition, a version can be proven which uses instead of .

f g ⇒ ⊢ ⟨skipf skipg FRightSkipChopImpSkipChopRule

Proof:

1
f g
Assump
2
(f g)
3
(f g) skipf skipg
4
skipf skipg
2, 3,MP

qed

f g f g FNextImpNextRule

Proof:

1
f g
Assump
2
skipf skipg
3
f ≡⟨skipf
4
g ≡⟨skipg
5
f g
3, 4,Prop

qed

f g f g FNextEqvNextRule

Proof:

1
f g
Assump
2
f g
1,Prop
3
f g
4
g f
1,Prop
5
g f
6
f g
3, 5,Prop

qed

f f f FDiamondEqvNowOrNextDiamond

Proof:

1
f ≡⟨skipf
2
skipf f skip ; skipf
3
skip ; skipf ≡⟨skip⟩⟨skipf
4
skipf ≡⟨skip⟩⟨skipf
5
f skipX
6
f f f
15,Prop

qed

f f FNowImpDiamond

Proof:

1
f f f
2
f f
1,Prop

qed

f f FNextDiamondImpDiamond

Proof:

1
f f f
2
f f
1,Prop

qed

f f BoxImpNow

Proof:

1
¬f ¬f
2
¬¬f f
1,Prop
3
f f
2, def. of 

qed

f f FBoxImpWeakNextBox

Proof:

1
¬¬¬f ¬f
2
¬¬¬f ¬f
3
¬f ¬f
4
¬¬¬f ¬f
2, 3,ImpChain
5
¬f ¬f
4, def. of 
6
¬¬f ¬¬f
5,Prop
7
f f
6, def. of ,

qed

Below is a proof of PTL Axiom A4:

f f f FBoxImpNowAndWeakNext

Proof:

1
f f
2
f f
3
f f f
1, 2,Prop

qed







2018-10-21
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