## Chandra-Toueg

### Safety properties

Below are the safety properties we want to verify, in negated form.
Exists only one Coordinator in the system:
∃ x,y. x≠y ∧ ( Coord[x] = True ∧ Coord[y] = True )
If 'c' is Coordinator, all other process with Id < c have already been Coordinator:
∃ x,y. x≠y ∧ ( Coord[x] = True ∧ ACoord[y] = False ∧ y < x )
A Coordinator can't have Id[] greater than his Identificator:
∃ x. ( Coord[x] = True ∧ Id_valid[x] = MValid ∧ x < Id[x] )
A process can't have Id[] greater than Coordinator's Identificator:
∃ x,y. x≠y ∧ ( Coord[x] = True ∧ Id_valid[y] = MValid ∧ x < Id[y] )
In the first Round a process can't have Id[] equals to the Coordinator's Identificator:
∃ x,y. x≠y ∧ ( Round = R1 ∧ Coord[x] = True ∧ Id_valid[y] = MValid ∧ Id[y] = x )
A correct process can't receive any Nack:
∃ x. ( Faulty[x] = False ∧ Nack[x] = True )
Coordinators are elected in order by Identificator:
∃ x,y. x≠y ∧ ( Coord[x] = True ∧ ACoord[y] = True ∧ x < y )
LEMMA 2.1:
∃ x,y,z. x≠y≠z ∧ ( State[x] = True ∧ Received[x] = True ∧ Coord[y] = True ∧
State[z] = False ∧ Faulty[z]= False ∧ Estimate[y] ≠ Estimate[z] )

LEMMA 2.2:
∃ x,y. x≠y ∧ ( Round ≠ R1 ∧ State[x] = False ∧ Faulty[x] = False ∧
ProcessReceivedEstimate[x] = True ∧ Coord[y] = True ∧ Estimate[x] ≠ Estimate[y] )

LEMMA 2.3:
∃ x,y. x≠y ∧ ( Coord[x] = True ∧ Faulty[x] = False ∧ Round = R6 ∧ State[y] = False ∧ Faulty[y] = False )
AGREEMENT - complete:
∃ x,y. x≠y ∧ ( State[x] = True ∧ Faulty[x] = False ∧ State[y] = True ∧ Faulty[y] = False ∧
DecisionValue[x] ≠ DecisionValue[y] )

-brab 2

### Inferred invariants

All invariants are shown in their negated form, where #1 and #2 are distinct existentially quantified variables.

```Received[#1] = True && Id_valid[#1] = M1

State[#1] = False && ACoord[#1] = True

Id_valid[#1] = M1 && ProcessReceivedEstimate[#1] = True

Round = R1 && Received[#1] = True

Round = R7 && Coord[#1] = True
```

You can find the list of all invariants that can be extracted from BRAB here (also in negated form), this collection being inductive.

### Search graph

The algorithm starts from the formula located at the bottom, inside a red octagon. Variables #1, #2, … that appear in the nodes are distinct skolem variables so we show a formula φ(#1, #2) as equivalent to ∃ z1, z2. z1 ≠ z2 ∧ φ(z1, z2). Plain black edges represent pre-image relations and are annotated by the transition instance that was considered. Black circles denote nodes that were obtained by pre-image computation and were not covered by already visited nodes. The nodes circled in gray are the one that were not useful because they were subsumed by formulas pointed by the gray dashed arrows. Approximations are shown in blue rectangles. Each approximation originates from the node that connects its rectangle with a bold dashed blue edge.