LTS Termination Proof

by T2Cert

Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

0: 2 − k_7_0 ≤ 02 − w_8_0 ≤ 0
1: 2 − k_7_0 ≤ 02 − w_8_0 ≤ 0
2: 2 − k_7_0 ≤ 02 − w_8_0 ≤ 0
3: TRUE
4: TRUE
5: TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
0 7 0: y_6_post + y_6_post ≤ 0y_6_posty_6_post ≤ 0y_6_0 + y_6_0 ≤ 0y_6_0y_6_0 ≤ 0x_5_post + x_5_post ≤ 0x_5_postx_5_post ≤ 0x_5_0 + x_5_0 ≤ 0x_5_0x_5_0 ≤ 0w_8_0 + w_8_0 ≤ 0w_8_0w_8_0 ≤ 0k_7_0 + k_7_0 ≤ 0k_7_0k_7_0 ≤ 0Result_4_post + Result_4_post ≤ 0Result_4_postResult_4_post ≤ 0Result_4_0 + Result_4_0 ≤ 0Result_4_0Result_4_0 ≤ 0
and for every transition t, a duplicate t is considered.

3 Transition Removal

We remove transitions 3, 4, 5, 6 using the following ranking functions, which are bounded by −13.

5: 0
4: 0
0: 0
1: 0
2: 0
3: 0
5: −5
4: −6
0: −7
1: −7
2: −7
0_var_snapshot: −7
0*: −7
3: −11

4 Location Addition

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

0* 10 0: y_6_post + y_6_post ≤ 0y_6_posty_6_post ≤ 0y_6_0 + y_6_0 ≤ 0y_6_0y_6_0 ≤ 0x_5_post + x_5_post ≤ 0x_5_postx_5_post ≤ 0x_5_0 + x_5_0 ≤ 0x_5_0x_5_0 ≤ 0w_8_0 + w_8_0 ≤ 0w_8_0w_8_0 ≤ 0k_7_0 + k_7_0 ≤ 0k_7_0k_7_0 ≤ 0Result_4_post + Result_4_post ≤ 0Result_4_postResult_4_post ≤ 0Result_4_0 + Result_4_0 ≤ 0Result_4_0Result_4_0 ≤ 0

5 Location Addition

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

0 8 0_var_snapshot: y_6_post + y_6_post ≤ 0y_6_posty_6_post ≤ 0y_6_0 + y_6_0 ≤ 0y_6_0y_6_0 ≤ 0x_5_post + x_5_post ≤ 0x_5_postx_5_post ≤ 0x_5_0 + x_5_0 ≤ 0x_5_0x_5_0 ≤ 0w_8_0 + w_8_0 ≤ 0w_8_0w_8_0 ≤ 0k_7_0 + k_7_0 ≤ 0k_7_0k_7_0 ≤ 0Result_4_post + Result_4_post ≤ 0Result_4_postResult_4_post ≤ 0Result_4_0 + Result_4_0 ≤ 0Result_4_0Result_4_0 ≤ 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 { 0, 1, 2, 0_var_snapshot, 0* }.

6.1.1 Transition Removal

We remove transition 0 using the following ranking functions, which are bounded by 7.

0: −1 − 5⋅x_5_0
1: 2 − 5⋅x_5_0
2: 1 − 5⋅x_5_0
0_var_snapshot: −2 − 5⋅x_5_0
0*: −5⋅x_5_0

6.1.2 Transition Removal

We remove transitions 8, 10, 1, 2 using the following ranking functions, which are bounded by −1.

0: 0
1: 1 + w_8_0
2: w_8_0
0_var_snapshot: −1
0*: −1 + w_8_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 7.

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