# [tlaplus] Re: Help with a TLAPS proof for a refinement involving records (and a Proof Decomposition bug)

Stephan wrote:

please feel free to post a bug report about the malfunctioning proof decomposition.

I do not see any malfunction of the proof decomposition command.  The command is expanding the
definition of  SV!IncreaseMaxBal(p, b).  The definition of IncreaseMaxBal in module SimpleVoting
uses the operator < (imported from the Naturals module), which is imported by the INSTANCE statement
as the operator SV!< .  Since the module containing the theorem also imports the Naturals module,
the operator SV!< happens to equal the operator < of the current module.  Please replace < by SV!<
in the proof and check that it still works.  If it doesn't, please report this as a TLAPS bug.

Because the definition of < is imported into the instantiated module from a standard module, and
that same definition is imported into the instantiating module from the same standard module, a
user would almost always prefer that the Decompose Proof command replace SV!< by < .  This should
be straightforward to implement and we would be grateful if someone would volunteer to do it.

Leslie

On Wednesday, August 14, 2019 at 9:26:22 PM UTC-7, Hengfeng Wei wrote:
Dear All,

I have some difficutlty in proving a refinement mapping involving records.

SimpleVoting.tla:

It maintains for each participant a maxBal which is a natural number.
In IncreaseMaxBal(p, b), maxBal[p] is increased to a larger value b.
---------------------------- MODULE SimpleVoting ----------------------------EXTENDS Naturals-----------------------------------------------------------------------------CONSTANT ParticipantVARIABLE maxBalTypeOK == maxBal \in [Participant -> Nat]-----------------------------------------------------------------------------Init == maxBal = [p \in Participant |-> 0]IncreaseMaxBal(p, b) ==  /\ maxBal[p] < b  /\ maxBal' = [maxBal EXCEPT ![p] = b]-----------------------------------------------------------------------------Next == \E p \in Participant, b \in Nat : IncreaseMaxBal(p, b)Spec == Init /\ [][Next]_maxBal=============================================================================

Record.tla:
It maintains a 2D array state. state[p][q] is the State of q from the view of p and a State is a record:
State == [maxBal : Nat, maxVBal : Nat].

In Prepare(p, b): state[p][p].maxBal is increased to a larger value b.
------------------------------- MODULE Record -------------------------------EXTENDS Naturals, TLAPS---------------------------------------------------------------------------CONSTANTS Participant  \* the set of partipantsVARIABLES state \* state[p][q]: the state of q \in Participant from the view of p \in Participant    State == [maxBal: Nat, maxVBal: Nat]TypeOK == state \in [Participant -> [Participant -> State]]---------------------------------------------------------------------------InitState == [maxBal |-> 0, maxVBal |-> 0]Init == state = [p \in Participant |-> [q \in Participant |-> InitState]] Prepare(p, b) ==    /\ state[p][p].maxBal < b    /\ state' = [state EXCEPT ![p][p].maxBal = b]---------------------------------------------------------------------------Next == \E p \in Participant, b \in Nat : Prepare(p, b)Spec == Init /\ [][Next]_state---------------------------------------------------------------------------

I want to show that Record refines SimpleVoting under the refinement mapping:

maxBal == [p \in Participant |-> state[p][p].maxBal]SV == INSTANCE SimpleVoting

However, step <4>2 fails (more details below). (In addition, the proof decomposition (Ctrl+G, Ctrl+D) of <3> generates a buggy formula "maxBal[p] SV!< b" in <4>1, which should be "maxBal[p] < b".)

THEOREM Spec => SV!Spec  <1>1. Init => SV!Init    BY DEF Init, SV!Init, maxBal, InitState  <1>2. [Next]_state => [SV!Next]_maxBal    <2>1. UNCHANGED state => UNCHANGED maxBal      BY DEF maxBal    <2>2. Next => SV!Next      <3> ASSUME NEW p \in Participant, NEW b \in Nat,                 Prepare(p, b)          PROVE  SV!IncreaseMaxBal(p, b) \* SV!Next        <4>1. maxBal[p] < b \* Wrong decomposition here: maxBal[p] SV!< b          BY DEF Prepare, maxBal        <4>2. maxBal' = [maxBal EXCEPT ![p] = b]          BY DEF Prepare, maxBal \* Failed here!!!        <4>3. QED          BY <4>1, <4>2 DEF SV!IncreaseMaxBal      <3>1. QED        BY DEF Next, SV!Next    <2>3. QED      BY <2>1, <2>2  <1>3. QED    BY <1>1, <1>2, PTL DEF SV!Spec, Spec

The obligation at <4>2 is as follows. Isn't the state' = [state EXCEPT ![p] = ...] in the assumption the same as the conclusion [p_1 \in Participant |-> state[p_1][p_1].maxBal]' ...? What is wrong with my proof? How should I proceed?

ASSUME NEW CONSTANT Participant,       NEW VARIABLE state,       NEW CONSTANT p \in Participant,       NEW CONSTANT b \in Nat,       /\ state[p][p].maxBal < b       /\ state'          = [state EXCEPT               ![p] = [state[p] EXCEPT                         ![p] = [state[p][p] EXCEPT !.maxBal = b]]] PROVE  [p_1 \in Participant |-> state[p_1][p_1].maxBal]'       = [[p_1 \in Participant |-> state[p_1][p_1].maxBal] EXCEPT ![p] = b]

Best regards,
hengxin

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