Thanks Stephan.
Yes, I also thought that I needed some axioms regarding removing and adding elements to finite sets. I didn't know about the FiniteSetsTheorem module though. I tried to define my own. Below is my current theorem. I'll look into the standard module now (but would still be interested in understanding why my axioms don't work.)
THEOREM ax1 == ASSUME NEW x \in Int PROVE IsFiniteSet({x})
THEOREM ax2 == ASSUME NEW x \in Int, NEW y \in Int PROVE IsFiniteSet({y-x})
THEOREM ax3 == ASSUME NEW S1, NEW S2,
/\ IsFiniteSet(S1)
/\ IsFiniteSet(S2)
PROVE /\ IsFiniteSet(S1 \ S2)
/\ IsFiniteSet(S1 \cup S2)
THEOREM Spec => []PartialCorrectness
<1>1. Init => SInv
BY InputOK DEF Init, SInv, TypeOK, PartialCorrectness
<1>2. SInv /\ [Next]_vars => SInv'
<2> SUFFICES ASSUME SInv /\ [Next]_vars
PROVE SInv'
OBVIOUS
<2>1. TypeOK'
<3>1. CASE Lbl_1
<4>1. (S \subseteq Nat \{0})'
BY <3>1 DEF SInv, TypeOK, Next, vars, Lbl_1
<4>2. (S # {})'
BY <3>1 DEF SetGCD, SInv, TypeOK, Next, vars, Lbl_1, SetMax, Divides
<4>3. IsFiniteSet(S)'
BY <3>1, ax1, ax2, ax3 DEF SInv, TypeOK, Next, vars, Lbl_1
<4>4. QED
BY <4>1, <4>2, <4>3 DEF TypeOK
<3>2. CASE pc = "Done" /\ UNCHANGED vars
BY <3>2 DEF PartialCorrectness, SInv, TypeOK, vars
<3>3. CASE UNCHANGED vars
BY <3>3 DEF Init, SetGCD, PartialCorrectness, SInv, TypeOK, Next, vars
<3>4. QED
BY <3>1, <3>2, <3>3 DEF Next
<2>2. (SetGCD(S) = SetGCD(T))'
BY InputOK DEF Init, SInv, TypeOK, Next, vars
<2>3. PartialCorrectness'
BY InputOK DEF Init, SInv, TypeOK, Next, vars
<2>4. QED
BY <2>1, <2>2, <2>3 DEF SInv
<1>3. SInv => PartialCorrectness
BY DEF SInv, PartialCorrectness
<1>4. QED
BY <1>1, <1>2, <1>3, PTL DEF Spec
On Friday, June 24, 2016 at 11:15:34 AM UTC-7, Stephan Merz wrote:
I haven't tried writing a formal proof of that algorithm, and it is not entirely clear to me where you are stuck.
The (type) invariant contains the conjunct
IsFiniteSet(S)
and you need to prove that this predicate is preserved by action Lbl_1. You'll need to use the standard module FiniteSetTheorems contained in the TLAPS distribution. (If you haven't done so yet, you should add the corresponding directory to the Toolbox search path.) In particular, the lemmas FS_AddElement and FS_RemoveElement will be useful.
Hope this helps,
Stephan