[Date Prev][Date Next][Thread Prev][Thread Next][Date Index][Thread Index]

Re: [tlaplus] Proving partial correctness of the SetGCD algorithm in the hyperbook

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})


                      /\ 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'


  <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


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,

On 24 Jun 2016, at 02:55, Jens Weber <jens...@xxxxxxxxx> wrote:

Hello TLA community,

I am trying to prove partial correctness of the SetGCD algorithm in the hyperbook - but I am not successful. Does anybody have a solution here? I am currently stuck on showing that S' is a finite set. 


You received this message because you are subscribed to the Google Groups "tlaplus" group.
To unsubscribe from this group and stop receiving emails from it, send an email to tlaplus+u...@googlegroups.com.
To post to this group, send email to tla...@xxxxxxxxxxxxxxxx.
Visit this group at https://groups.google.com/group/tlaplus.
For more options, visit https://groups.google.com/d/optout.