LTS Termination Proof

by T2Cert

Input

Integer Transition System

Initial Location: 5

Transitions: (pre-variables and post-variables)

0	0	1:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ 1 − arg2 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ 1 − arg1P ≤ 0 ∧ − arg2P ≤ 0 ∧ arg2P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0 ∧ − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 ∧ − x28 + x28 ≤ 0 ∧ x28 − x28 ≤ 0 ∧ − x19 + x19 ≤ 0 ∧ x19 − x19 ≤ 0 ∧ − x11 + x11 ≤ 0 ∧ x11 − x11 ≤ 0
0	1	1:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − arg2P ≤ 0 ∧ 1 − arg2 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ 1 − arg1P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0 ∧ − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 ∧ − x28 + x28 ≤ 0 ∧ x28 − x28 ≤ 0 ∧ − x19 + x19 ≤ 0 ∧ x19 − x19 ≤ 0 ∧ − x11 + x11 ≤ 0 ∧ x11 − x11 ≤ 0
1	2	2:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ 2 − x7 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ 1 − arg1P ≤ 0 ∧ − arg3P ≤ 0 ∧ arg3P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0 ∧ − x28 + x28 ≤ 0 ∧ x28 − x28 ≤ 0 ∧ − x19 + x19 ≤ 0 ∧ x19 − x19 ≤ 0 ∧ − x11 + x11 ≤ 0 ∧ x11 − x11 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0
1	3	2:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 2 − x11 ≤ 0 ∧ − arg3P ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ 1 − arg1P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0 ∧ − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 ∧ − x28 + x28 ≤ 0 ∧ x28 − x28 ≤ 0 ∧ − x19 + x19 ≤ 0 ∧ x19 − x19 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0
0	4	3:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ − arg2 ≤ 0 ∧ arg2 ≤ 0 ∧ − arg1P ≤ 0 ∧ arg1P ≤ 0 ∧ − arg2P ≤ 0 ∧ arg2P ≤ 0 ∧ − arg3P ≤ 0 ∧ arg3P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0 ∧ − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 ∧ − x28 + x28 ≤ 0 ∧ x28 − x28 ≤ 0 ∧ − x19 + x19 ≤ 0 ∧ x19 − x19 ≤ 0 ∧ − x11 + x11 ≤ 0 ∧ x11 − x11 ≤ 0
1	5	3:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ − arg1P ≤ 0 ∧ arg1P ≤ 0 ∧ − arg3P ≤ 0 ∧ arg3P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0 ∧ − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 ∧ − x28 + x28 ≤ 0 ∧ x28 − x28 ≤ 0 ∧ − x19 + x19 ≤ 0 ∧ x19 − x19 ≤ 0 ∧ − x11 + x11 ≤ 0 ∧ x11 − x11 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0
2	6	3:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ 2 − x19 ≤ 0 ∧ − arg1P + arg3 ≤ 0 ∧ arg1P − arg3 ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 ∧ − x28 + x28 ≤ 0 ∧ x28 − x28 ≤ 0 ∧ − x11 + x11 ≤ 0 ∧ x11 − x11 ≤ 0 ∧ − arg3P + arg3P ≤ 0 ∧ arg3P − arg3P ≤ 0 ∧ − arg3 + arg3 ≤ 0 ∧ arg3 − arg3 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0
3	7	4:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 2 − arg2 ≤ 0 ∧ 2 − arg1 ≤ 0 ∧ 1 − arg2 + x28 ≤ 0 ∧ arg1 − arg2 ≤ 0 ∧ arg1 − arg3 ≤ 0 ∧ − arg1 + arg3 ≤ 0 ∧ arg1 − arg3P ≤ 0 ∧ − arg1 + arg3P ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0 ∧ − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 ∧ − x19 + x19 ≤ 0 ∧ x19 − x19 ≤ 0 ∧ − x11 + x11 ≤ 0 ∧ x11 − x11 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0
4	8	3:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 2 − arg2 ≤ 0 ∧ 2 − arg1 ≤ 0 ∧ arg1 − arg2 ≤ 0 ∧ 1 + arg2P − arg2 ≤ 0 ∧ arg1 − arg3 ≤ 0 ∧ − arg1 + arg3 ≤ 0 ∧ arg1 − arg3P ≤ 0 ∧ − arg1 + arg3P ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0 ∧ − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 ∧ − x28 + x28 ≤ 0 ∧ x28 − x28 ≤ 0 ∧ − x19 + x19 ≤ 0 ∧ x19 − x19 ≤ 0 ∧ − x11 + x11 ≤ 0 ∧ x11 − x11 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0
5	9	0:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0 ∧ − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 ∧ − x28 + x28 ≤ 0 ∧ x28 − x28 ≤ 0 ∧ − x19 + x19 ≤ 0 ∧ x19 − x19 ≤ 0 ∧ − x11 + x11 ≤ 0 ∧ x11 − x11 ≤ 0

Proof

1 Invariant Updates

The following invariants are asserted.

0:	TRUE
1:	1 − arg1P ≤ 0 ∧ − arg2P ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ − arg2 ≤ 0
2:	1 − arg1P ≤ 0 ∧ − arg2P ≤ 0 ∧ − arg3P ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ − arg2 ≤ 0 ∧ − arg3 ≤ 0
3:	TRUE
4:	2 − arg1 ≤ 0 ∧ 2 − arg2 ≤ 0
5:	TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

nodes (location) invariant:

0	(0)	TRUE
1	(1)	1 − arg1P ≤ 0 ∧ − arg2P ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ − arg2 ≤ 0
2	(2)	1 − arg1P ≤ 0 ∧ − arg2P ≤ 0 ∧ − arg3P ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ − arg2 ≤ 0 ∧ − arg3 ≤ 0
3	(3)	TRUE
4	(4)	2 − arg1 ≤ 0 ∧ 2 − arg2 ≤ 0
5	(5)	TRUE

initial node: 5
cover edges:
transition edges:

0 0 1

0 1 1

0 4 3

1 2 2

1 3 2

1 5 3

2 6 3

3 7 4

4 8 3

5 9 0

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

− x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 ∧ − x28 + x28 ≤ 0 ∧ x28 − x28 ≤ 0 ∧ − x19 + x19 ≤ 0 ∧ x19 − x19 ≤ 0 ∧ − x11 + x11 ≤ 0 ∧ x11 − x11 ≤ 0 ∧ − arg3P + arg3P ≤ 0 ∧ arg3P − arg3P ≤ 0 ∧ − arg3 + arg3 ≤ 0 ∧ arg3 − arg3 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0

and for every transition t, a duplicate t is considered.

3 Transition Removal

We remove transitions 0, 1, 2, 3, 4, 5, 6, 9 using the following ranking functions, which are bounded by −15.

5:	0
0:	0
1:	0
2:	0
3:	0
4:	0
5:	−6
0:	−7
1:	−8
2:	−9
3:	−10
4:	−10
3_var_snapshot:	−10
3*:	−10

4 Location Addition

The following skip-transition is inserted and corresponding redirections w.r.t. the old location are performed.

3* 13 3: − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 ∧ − x28 + x28 ≤ 0 ∧ x28 − x28 ≤ 0 ∧ − x19 + x19 ≤ 0 ∧ x19 − x19 ≤ 0 ∧ − x11 + x11 ≤ 0 ∧ x11 − x11 ≤ 0 ∧ − arg3P + arg3P ≤ 0 ∧ arg3P − arg3P ≤ 0 ∧ − arg3 + arg3 ≤ 0 ∧ arg3 − arg3 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0

5 Location Addition

The following skip-transition is inserted and corresponding redirections w.r.t. the old location are performed.

3 11 3_var_snapshot: − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 ∧ − x28 + x28 ≤ 0 ∧ x28 − x28 ≤ 0 ∧ − x19 + x19 ≤ 0 ∧ x19 − x19 ≤ 0 ∧ − x11 + x11 ≤ 0 ∧ x11 − x11 ≤ 0 ∧ − arg3P + arg3P ≤ 0 ∧ arg3P − arg3P ≤ 0 ∧ − arg3 + arg3 ≤ 0 ∧ arg3 − arg3 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0

6 SCC Decomposition

We consider subproblems for each of the 1 SCC(s) of the program graph.

6.1 SCC Subproblem 1/1

Here we consider the SCC { 3, 4, 3_var_snapshot, 3* }.

6.1.1 Transition Removal

We remove transition 7 using the following ranking functions, which are bounded by 5.

3:	−1 + 4⋅arg2
4:	−3 + 4⋅arg2
3_var_snapshot:	−2 + 4⋅arg2
3*:	4⋅arg2

6.1.2 Transition Removal

We remove transitions 11, 13, 8 using the following ranking functions, which are bounded by −2.

3:	−1
4:	1
3_var_snapshot:	−2
3*:	0

6.1.3 Splitting Cut-Point Transitions

We consider 1 subproblems corresponding to sets of cut-point transitions as follows.

6.1.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 10.

6.1.3.1.1 Splitting Cut-Point Transitions

There remain no cut-point transition to consider. Hence the cooperation termination is trivial.

Tool configuration

T2Cert

version: 1.0