LTS Termination Proof

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Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

0: TRUE
1: arg1P ≤ 0arg1 ≤ 0
2: arg1P ≤ 0arg1 ≤ 0
3: TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
1 5 1: arg3P + 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, 4 using the following ranking functions, which are bounded by −11.

3: 0
0: 0
1: 0
2: 0
3: −4
0: −5
1: −6
2: −6
1_var_snapshot: −6
1*: −6

4 Location Addition

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

1* 8 1: arg3P + 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 6 1_var_snapshot: arg3P + arg3P ≤ 0arg3Parg3P ≤ 0arg3 + arg3 ≤ 0arg3arg3 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 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 { 1, 2, 1_var_snapshot, 1* }.

6.1.1 Transition Removal

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

1: 0
2: arg2arg3
1_var_snapshot: 0
1*: 0

6.1.2 Splitting Cut-Point Transitions

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

6.1.2.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 5.

6.1.2.1.1 Fresh Variable Addition

The new variable __snapshot_1_arg3P is introduced. The transition formulas are extended as follows:

6: __snapshot_1_arg3Parg3Parg3P__snapshot_1_arg3P
8: __snapshot_1_arg3P__snapshot_1_arg3P__snapshot_1_arg3P__snapshot_1_arg3P
2: __snapshot_1_arg3P__snapshot_1_arg3P__snapshot_1_arg3P__snapshot_1_arg3P
3: __snapshot_1_arg3P__snapshot_1_arg3P__snapshot_1_arg3P__snapshot_1_arg3P

6.1.2.1.2 Fresh Variable Addition

The new variable __snapshot_1_arg3 is introduced. The transition formulas are extended as follows:

6: __snapshot_1_arg3arg3arg3__snapshot_1_arg3
8: __snapshot_1_arg3__snapshot_1_arg3__snapshot_1_arg3__snapshot_1_arg3
2: __snapshot_1_arg3__snapshot_1_arg3__snapshot_1_arg3__snapshot_1_arg3
3: __snapshot_1_arg3__snapshot_1_arg3__snapshot_1_arg3__snapshot_1_arg3

6.1.2.1.3 Fresh Variable Addition

The new variable __snapshot_1_arg2P is introduced. The transition formulas are extended as follows:

6: __snapshot_1_arg2Parg2Parg2P__snapshot_1_arg2P
8: __snapshot_1_arg2P__snapshot_1_arg2P__snapshot_1_arg2P__snapshot_1_arg2P
2: __snapshot_1_arg2P__snapshot_1_arg2P__snapshot_1_arg2P__snapshot_1_arg2P
3: __snapshot_1_arg2P__snapshot_1_arg2P__snapshot_1_arg2P__snapshot_1_arg2P

6.1.2.1.4 Fresh Variable Addition

The new variable __snapshot_1_arg2 is introduced. The transition formulas are extended as follows:

6: __snapshot_1_arg2arg2arg2__snapshot_1_arg2
8: __snapshot_1_arg2__snapshot_1_arg2__snapshot_1_arg2__snapshot_1_arg2
2: __snapshot_1_arg2__snapshot_1_arg2__snapshot_1_arg2__snapshot_1_arg2
3: __snapshot_1_arg2__snapshot_1_arg2__snapshot_1_arg2__snapshot_1_arg2

6.1.2.1.5 Fresh Variable Addition

The new variable __snapshot_1_arg1P is introduced. The transition formulas are extended as follows:

6: __snapshot_1_arg1Parg1Parg1P__snapshot_1_arg1P
8: __snapshot_1_arg1P__snapshot_1_arg1P__snapshot_1_arg1P__snapshot_1_arg1P
2: __snapshot_1_arg1P__snapshot_1_arg1P__snapshot_1_arg1P__snapshot_1_arg1P
3: __snapshot_1_arg1P__snapshot_1_arg1P__snapshot_1_arg1P__snapshot_1_arg1P

6.1.2.1.6 Fresh Variable Addition

The new variable __snapshot_1_arg1 is introduced. The transition formulas are extended as follows:

6: __snapshot_1_arg1arg1arg1__snapshot_1_arg1
8: __snapshot_1_arg1__snapshot_1_arg1__snapshot_1_arg1__snapshot_1_arg1
2: __snapshot_1_arg1__snapshot_1_arg1__snapshot_1_arg1__snapshot_1_arg1
3: __snapshot_1_arg1__snapshot_1_arg1__snapshot_1_arg1__snapshot_1_arg1

6.1.2.1.7 Invariant Updates

The following invariants are asserted.

0: TRUE
1: arg2arg2P ≤ 0−1 + arg1arg2P ≤ 0arg1P ≤ 0arg1 ≤ 0
2: arg1P ≤ 0arg1 ≤ 0
3: TRUE
1: arg2arg2P ≤ 0−1 + arg1arg2P ≤ 0arg1P ≤ 0arg1 ≤ 0__snapshot_1_arg2P + __snapshot_1_arg3 ≤ 0arg2arg2P ≤ 01 − __snapshot_1_arg2P + __snapshot_1_arg3 + arg2Parg3 ≤ 0−1 + arg1arg2P ≤ 0arg1P ≤ 0arg1 ≤ 0
2: __snapshot_1_arg2P + __snapshot_1_arg3 ≤ 0−1 − __snapshot_1_arg2P + __snapshot_1_arg3 + arg1arg2 ≤ 0arg1P ≤ 0arg1 ≤ 0
1_var_snapshot: −1 − __snapshot_1_arg2P + __snapshot_1_arg3 + arg1arg3 ≤ 0__snapshot_1_arg2P + __snapshot_1_arg3 + arg2arg3 ≤ 0arg1P ≤ 0arg1 ≤ 0
1*: __snapshot_1_arg2P + __snapshot_1_arg3 ≤ 0arg2arg2P ≤ 01 − __snapshot_1_arg2P + __snapshot_1_arg3 + arg2Parg3 ≤ 0−1 + arg1arg2P ≤ 0arg1P ≤ 0arg1 ≤ 0

The invariants are proved as follows.

IMPACT Invariant Proof

6.1.2.1.8 Transition Removal

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

1: arg2Parg3
2: __snapshot_1_arg2P__snapshot_1_arg3
1_var_snapshot: __snapshot_1_arg2P__snapshot_1_arg3
1*: __snapshot_1_arg2P__snapshot_1_arg3

6.1.2.1.9 Transition Removal

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

1: −1
1_var_snapshot: −2
2: −3
1*: −4

6.1.2.1.10 Splitting Cut-Point Transitions

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

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