LTS Termination Proof

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Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

0: TRUE
1: TRUE
2: 1 − arg1P ≤ 01 − arg2P ≤ 0arg3P ≤ 0arg3P ≤ 01 − arg1 ≤ 01 − arg2 ≤ 0arg3 ≤ 0arg3 ≤ 0
3: arg1P ≤ 0arg2P ≤ 0arg1 ≤ 0arg2 ≤ 01 − x15 ≤ 0
4: TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
1 9 1: x7 + x7 ≤ 0x7x7 ≤ 0x15 + x15 ≤ 0x15x15 ≤ 0x12 + x12 ≤ 0x12x12 ≤ 0arg3P + arg3P ≤ 0arg3Parg3P ≤ 0arg3 + arg3 ≤ 0arg3arg3 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0
3 16 3: x7 + x7 ≤ 0x7x7 ≤ 0x15 + x15 ≤ 0x15x15 ≤ 0x12 + x12 ≤ 0x12x12 ≤ 0arg3P + arg3P ≤ 0arg3Parg3P ≤ 0arg3 + arg3 ≤ 0arg3arg3 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0
and for every transition t, a duplicate t is considered.

3 Transition Removal

We remove transitions 0, 1, 2, 4, 8 using the following ranking functions, which are bounded by −17.

4: 0
0: 0
1: 0
2: 0
3: 0
4: −6
0: −7
1: −8
1_var_snapshot: −8
1*: −8
2: −11
3: −12
3_var_snapshot: −12
3*: −12

4 Location Addition

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

1* 12 1: x7 + x7 ≤ 0x7x7 ≤ 0x15 + x15 ≤ 0x15x15 ≤ 0x12 + x12 ≤ 0x12x12 ≤ 0arg3P + arg3P ≤ 0arg3Parg3P ≤ 0arg3 + arg3 ≤ 0arg3arg3 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

5 Location Addition

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

1 10 1_var_snapshot: x7 + x7 ≤ 0x7x7 ≤ 0x15 + x15 ≤ 0x15x15 ≤ 0x12 + x12 ≤ 0x12x12 ≤ 0arg3P + arg3P ≤ 0arg3Parg3P ≤ 0arg3 + arg3 ≤ 0arg3arg3 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

6 Location Addition

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

3* 19 3: x7 + x7 ≤ 0x7x7 ≤ 0x15 + x15 ≤ 0x15x15 ≤ 0x12 + x12 ≤ 0x12x12 ≤ 0arg3P + arg3P ≤ 0arg3Parg3P ≤ 0arg3 + arg3 ≤ 0arg3arg3 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

7 Location Addition

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

3 17 3_var_snapshot: x7 + x7 ≤ 0x7x7 ≤ 0x15 + x15 ≤ 0x15x15 ≤ 0x12 + x12 ≤ 0x12x12 ≤ 0arg3P + arg3P ≤ 0arg3Parg3P ≤ 0arg3 + arg3 ≤ 0arg3arg3 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

8 SCC Decomposition

We consider subproblems for each of the 2 SCC(s) of the program graph.

8.1 SCC Subproblem 1/2

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

8.1.1 Transition Removal

We remove transition 3 using the following ranking functions, which are bounded by −1.

1: 1 + 3⋅arg1
1_var_snapshot: 3⋅arg1
1*: 2 + 3⋅arg1

8.1.2 Transition Removal

We remove transition 12 using the following ranking functions, which are bounded by −2.

1: −1
1_var_snapshot: −2
1*: 0

8.1.3 Transition Removal

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

1: 0
1_var_snapshot: −1
1*: 0

8.1.4 Splitting Cut-Point Transitions

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

8.1.4.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 9.

8.1.4.1.1 Splitting Cut-Point Transitions

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

8.2 SCC Subproblem 2/2

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

8.2.1 Transition Removal

We remove transition 6 using the following ranking functions, which are bounded by −1.

3: arg2
3_var_snapshot: arg2
3*: arg2

8.2.2 Transition Removal

We remove transition 5 using the following ranking functions, which are bounded by −1.

3: arg1
3_var_snapshot: arg1
3*: arg1

8.2.3 Transition Removal

We remove transition 7 using the following ranking functions, which are bounded by −1.

3: −3⋅arg3
3_var_snapshot: −3⋅arg3
3*: 1 − 3⋅arg3

8.2.4 Transition Removal

We remove transitions 17, 19 using the following ranking functions, which are bounded by −1.

3: 0
3_var_snapshot: −1
3*: x15

8.2.5 Splitting Cut-Point Transitions

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

8.2.5.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 16.

8.2.5.1.1 Splitting Cut-Point Transitions

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

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