LTS Termination Proof

by T2Cert

Input

Integer Transition System

Initial Location: 4

Transitions: (pre-variables and post-variables)

0	0	1:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 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
1	1	1:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − arg2 ≤ 0 ∧ − arg1 ≤ 0 ∧ −1000 + arg2 ≤ 0 ∧ arg2 − arg3 ≤ 0 ∧ − arg2 + arg3 ≤ 0 ∧ − arg1P + arg1 + arg2 ≤ 0 ∧ arg1P − arg1 − arg2 ≤ 0 ∧ 1 − arg2P + arg2 ≤ 0 ∧ −1 + arg2P − arg2 ≤ 0 ∧ 1 + arg2 − arg3P ≤ 0 ∧ −1 − arg2 + arg3P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0
1	2	2:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1001 − arg2 ≤ 0 ∧ arg2 − arg3 ≤ 0 ∧ − arg2 + arg3 ≤ 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
2	3	2:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − arg2 ≤ 0 ∧ − arg1 ≤ 0 ∧ −1000 + arg2 ≤ 0 ∧ arg2 − arg3 ≤ 0 ∧ − arg2 + arg3 ≤ 0 ∧ − arg1P + arg1 + arg2 ≤ 0 ∧ arg1P − arg1 − arg2 ≤ 0 ∧ 2 − arg2P + arg2 ≤ 0 ∧ −2 + arg2P − arg2 ≤ 0 ∧ 2 + arg2 − arg3P ≤ 0 ∧ −2 − arg2 + arg3P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0
2	4	3:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1001 − arg2 ≤ 0 ∧ arg2 − arg3 ≤ 0 ∧ − arg2 + arg3 ≤ 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
3	5	3:		0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − arg2 ≤ 0 ∧ − arg1 ≤ 0 ∧ −1000 + arg2 ≤ 0 ∧ arg2 − arg3 ≤ 0 ∧ − arg2 + arg3 ≤ 0 ∧ − arg1P + arg1 + arg2 ≤ 0 ∧ arg1P − arg1 − arg2 ≤ 0 ∧ 3 − arg2P + arg2 ≤ 0 ∧ −3 + arg2P − arg2 ≤ 0 ∧ 3 + arg2 − arg3P ≤ 0 ∧ −3 − arg2 + arg3P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0
4	6	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

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

1	7	1:	− 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
2	14	2:	− 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
3	21	3:	− 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.

2 Transition Removal

We remove transitions 0, 2, 4, 6 using the following ranking functions, which are bounded by −19.

4:	0
0:	0
1:	0
2:	0
3:	0
4:	−6
0:	−7
1:	−8
1_var_snapshot:	−8
1*:	−8
2:	−11
2_var_snapshot:	−11
2*:	−11
3:	−14
3_var_snapshot:	−14
3*:	−14

3 Location Addition

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

1* 10 1: − 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

4 Location Addition

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

1 8 1_var_snapshot: − 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.

2* 17 2: − 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 Location Addition

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

2 15 2_var_snapshot: − 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

7 Location Addition

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

3* 24 3: − 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

8 Location Addition

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

3 22 3_var_snapshot: − 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

9 SCC Decomposition

We consider subproblems for each of the 3 SCC(s) of the program graph.

9.1 SCC Subproblem 1/3

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

9.1.1 Transition Removal

We remove transition 5 using the following ranking functions, which are bounded by −1000003.

3:	−1 − 1000⋅arg2
3_var_snapshot:	−2 − 1000⋅arg2
3*:	−1000⋅arg2

9.1.2 Transition Removal

We remove transition 22 using the following ranking functions, which are bounded by −1.

3:	0
3_var_snapshot:	−1
3*:	1

9.1.3 Transition Removal

We remove transition 24 using the following ranking functions, which are bounded by −1.

3:	−1
3_var_snapshot:	0
3*:	0

9.1.4 Splitting Cut-Point Transitions

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

9.1.4.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 21.

9.1.4.1.1 Splitting Cut-Point Transitions

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

9.2 SCC Subproblem 2/3

Here we consider the SCC { 2, 2_var_snapshot, 2* }.

9.2.1 Transition Removal

We remove transition 3 using the following ranking functions, which are bounded by −1000003.

2:	−1 − 500⋅arg2 − 500⋅arg3
2_var_snapshot:	−2 − 500⋅arg2 − 500⋅arg3
2*:	−500⋅arg2 − 500⋅arg3

9.2.2 Transition Removal

We remove transition 15 using the following ranking functions, which are bounded by −1.

2:	0
2_var_snapshot:	−1
2*:	1

9.2.3 Transition Removal

We remove transition 17 using the following ranking functions, which are bounded by −1.

2:	−1
2_var_snapshot:	0
2*:	0

9.2.4 Splitting Cut-Point Transitions

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

9.2.4.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 14.

9.2.4.1.1 Splitting Cut-Point Transitions

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

9.3 SCC Subproblem 3/3

Here we consider the SCC { 1, 1_var_snapshot, 1* }.

9.3.1 Transition Removal

We remove transition 1 using the following ranking functions, which are bounded by −2002.

1:	−1 − 2⋅arg3
1_var_snapshot:	−1 − 2⋅arg3
1*:	−2⋅arg3

9.3.2 Transition Removal

We remove transition 8 using the following ranking functions, which are bounded by −1.

1:	0
1_var_snapshot:	−1
1*:	1

9.3.3 Transition Removal

We remove transition 10 using the following ranking functions, which are bounded by −1.

1:	−1
1_var_snapshot:	0
1*:	0

9.3.4 Splitting Cut-Point Transitions

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

9.3.4.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 7.

9.3.4.1.1 Splitting Cut-Point Transitions

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

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