LTS Termination Proof

by T2Cert

Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

0: TRUE
1: 1 − arg1P ≤ 0arg2P ≤ 01 − arg1 ≤ 0arg2 ≤ 0
2: 1 − arg1P ≤ 0arg2P ≤ 0arg3P ≤ 01 − arg1 ≤ 0arg2 ≤ 0arg3 ≤ 0
3: TRUE
4: 2 − arg1 ≤ 02 − arg2 ≤ 0
5: TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
3 10 3: x7 + x7 ≤ 0x7x7 ≤ 0x28 + x28 ≤ 0x28x28 ≤ 0x19 + x19 ≤ 0x19x19 ≤ 0x11 + x11 ≤ 0x11x11 ≤ 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, 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 ≤ 0x7x7 ≤ 0x28 + x28 ≤ 0x28x28 ≤ 0x19 + x19 ≤ 0x19x19 ≤ 0x11 + x11 ≤ 0x11x11 ≤ 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.

3 11 3_var_snapshot: x7 + x7 ≤ 0x7x7 ≤ 0x28 + x28 ≤ 0x28x28 ≤ 0x19 + x19 ≤ 0x19x19 ≤ 0x11 + x11 ≤ 0x11x11 ≤ 0arg3P + 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 { 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