LTS Termination Proof

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Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

0: TRUE
1: 1 − arg1 ≤ 01 − arg2 ≤ 0
2: TRUE
3: TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
2 6 2: x9 + x9 ≤ 0x9x9 ≤ 0x22 + x22 ≤ 0x22x22 ≤ 0x21 + x21 ≤ 0x21x21 ≤ 0x18 + x18 ≤ 0x18x18 ≤ 0x17 + x17 ≤ 0x17x17 ≤ 0x14 + x14 ≤ 0x14x14 ≤ 0x13 + x13 ≤ 0x13x13 ≤ 0x10 + x10 ≤ 0x10x10 ≤ 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, 3, 5 using the following ranking functions, which are bounded by −13.

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

4 Location Addition

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

2* 9 2: x9 + x9 ≤ 0x9x9 ≤ 0x22 + x22 ≤ 0x22x22 ≤ 0x21 + x21 ≤ 0x21x21 ≤ 0x18 + x18 ≤ 0x18x18 ≤ 0x17 + x17 ≤ 0x17x17 ≤ 0x14 + x14 ≤ 0x14x14 ≤ 0x13 + x13 ≤ 0x13x13 ≤ 0x10 + x10 ≤ 0x10x10 ≤ 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.

2 7 2_var_snapshot: x9 + x9 ≤ 0x9x9 ≤ 0x22 + x22 ≤ 0x22x22 ≤ 0x21 + x21 ≤ 0x21x21 ≤ 0x18 + x18 ≤ 0x18x18 ≤ 0x17 + x17 ≤ 0x17x17 ≤ 0x14 + x14 ≤ 0x14x14 ≤ 0x13 + x13 ≤ 0x13x13 ≤ 0x10 + x10 ≤ 0x10x10 ≤ 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 { 2, 2_var_snapshot, 2* }.

6.1.1 Splitting Cut-Point Transitions

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

6.1.1.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 6.

6.1.1.1.1 Invariant Updates

The following invariants are asserted.

0: TRUE
1: 1 − arg1 ≤ 01 − arg2 ≤ 0
2: 1 − arg1 + arg2 ≤ 01 + arg1arg2 ≤ 01 ≤ 0
3: TRUE
2: 1 + arg1arg2 ≤ 01 − arg1 + arg2 ≤ 0
2_var_snapshot: 1 + arg1arg2 ≤ 01 − arg1 + arg2 ≤ 0
2*: 1 ≤ 0

The invariants are proved as follows.

IMPACT Invariant Proof

6.1.1.1.2 Transition Removal

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

2: −1
2_var_snapshot: −2
2*: −3

6.1.1.1.3 Splitting Cut-Point Transitions

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

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