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# [tlaplus] meaning of the lead to (~>) operator

• From: seup yun <syun4541@xxxxxxxxx>
• Date: Sat, 10 Apr 2021 09:55:45 -0700 (PDT)
• Ironport-hdrordr: A9a23:kUjvM65Q1yJ+pRCqtAPXwA7XdLJzesId70hD6mlaTxtJfsuE0/2/hfhz726XtB89UGwt8OruBICuWnXZnKQa3aA0HZOPGDbro3GpKoYK1/qA/xTFFzfl/uBQkYdMGpIeNPTKAVJ3jdn37WCDer5K/PC87Kuqie3Cpk0BcShWbchbhTtRO0K+Gk1ySBIuP/oEPavZ3+t/4xy6eXoWacynBn4KG9LIvN3TifvdDSI7Ow==

Hi,

I'm trying to specify "If some operation does not happens, then some condition eventually becomes true"

full specification is following:
-------------------------------- MODULE test --------------------------------

EXTENDS Integers

VARIABLE x,y

Init ==
/\ x = 0
/\ y = 0

IncX ==
/\ x' = IF x < 5 THEN (x + 1) ELSE 5
/\ UNCHANGED <<y>>

IncY ==
/\ x > 0
/\ y' = IF y < 5 THEN (y + 1) ELSE 5
/\ UNCHANGED <<x>>

SetX ==
/\ x' = -1
/\ UNCHANGED <<y>>

Next ==
\/ IncX
\/ IncY
\/ SetX

Invariant1 ==
/\ x < 6
/\ y < 6

temporalFormula1 ==  (x > 0) ~> (y = 5)
=============================================================================

and behavior spec is:
Init /\ [][Next]_<<x,y>> /\ WF_x(IncX) /\ WF_x(SetX) /\ WF_<<x,y>>(IncY)

my question is, why temporalFormula1 is violated?

I know that (x > 0) ~> (y = 5) means:
((x > 0) => <>(y = 5)) which is
\A n \in Nat. \sigma^{+n} ⊨ (x > 0) => ¬□¬(y = 5) which is
\A n \in Nat. \sigma^{+n} ⊨ (x > 0) => \sigma^{+n} ⊨ \E m \in Nat. \sigma{+m} ⊨ (y = 5) which is
\A n \in Nat. \sigma^{+n} ⊨ (x > 0) => \E m \in Nat. \sigma^{+n+m} ⊨ (y = 5)

so, i believed that error-trace should not contains the state that x is -1 but it does.

I even tried the formula (FALSE) ~> (y = 5) but it still fails with similar error trace. what's my mistake here?

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