LTS Termination Proof

by T2Cert

Input

Integer Transition System

Initial Location: 5

Transitions: (pre-variables and post-variables)

0	0	1:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ i_13_1 − s_17_0 ≤ 0 ∧ − i_13_1 + s_17_0 ≤ 0 ∧ j_15_post − s_16_0 ≤ 0 ∧ − j_15_post + s_16_0 ≤ 0 ∧ i_13_post − j_15_post ≤ 0 ∧ − i_13_post + j_15_post ≤ 0 ∧ i_13_0 − i_13_post ≤ 0 ∧ − i_13_0 + i_13_post ≤ 0 ∧ j_15_0 − j_15_post ≤ 0 ∧ − j_15_0 + j_15_post ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − s_17_0 + s_17_0 ≤ 0 ∧ s_17_0 − s_17_0 ≤ 0 ∧ − s_16_0 + s_16_0 ≤ 0 ∧ s_16_0 − s_16_0 ≤ 0 ∧ − rt_11_post + rt_11_post ≤ 0 ∧ rt_11_post − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_0 ≤ 0 ∧ rt_11_0 − rt_11_0 ≤ 0 ∧ − ___const_500_0 + ___const_500_0 ≤ 0 ∧ ___const_500_0 − ___const_500_0 ≤ 0
2	1	3:	0 ≤ 0 ∧ 0 ≤ 0 ∧ ___const_500_0 − i_13_0 ≤ 0 ∧ rt_11_post − st_14_0 ≤ 0 ∧ − rt_11_post + st_14_0 ≤ 0 ∧ rt_11_0 − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_post ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − s_17_0 + s_17_0 ≤ 0 ∧ s_17_0 − s_17_0 ≤ 0 ∧ − s_16_0 + s_16_0 ≤ 0 ∧ s_16_0 − s_16_0 ≤ 0 ∧ − j_15_post + j_15_post ≤ 0 ∧ j_15_post − j_15_post ≤ 0 ∧ − j_15_0 + j_15_0 ≤ 0 ∧ j_15_0 − j_15_0 ≤ 0 ∧ − i_13_post + i_13_post ≤ 0 ∧ i_13_post − i_13_post ≤ 0 ∧ − i_13_1 + i_13_1 ≤ 0 ∧ i_13_1 − i_13_1 ≤ 0 ∧ − i_13_0 + i_13_0 ≤ 0 ∧ i_13_0 − i_13_0 ≤ 0 ∧ − ___const_500_0 + ___const_500_0 ≤ 0 ∧ ___const_500_0 − ___const_500_0 ≤ 0
2	2	4:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − ___const_500_0 + i_13_0 ≤ 0 ∧ −1 − i_13_0 + i_13_post ≤ 0 ∧ 1 + i_13_0 − i_13_post ≤ 0 ∧ i_13_0 − i_13_post ≤ 0 ∧ − i_13_0 + i_13_post ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − s_17_0 + s_17_0 ≤ 0 ∧ s_17_0 − s_17_0 ≤ 0 ∧ − s_16_0 + s_16_0 ≤ 0 ∧ s_16_0 − s_16_0 ≤ 0 ∧ − rt_11_post + rt_11_post ≤ 0 ∧ rt_11_post − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_0 ≤ 0 ∧ rt_11_0 − rt_11_0 ≤ 0 ∧ − j_15_post + j_15_post ≤ 0 ∧ j_15_post − j_15_post ≤ 0 ∧ − j_15_0 + j_15_0 ≤ 0 ∧ j_15_0 − j_15_0 ≤ 0 ∧ − i_13_1 + i_13_1 ≤ 0 ∧ i_13_1 − i_13_1 ≤ 0 ∧ − ___const_500_0 + ___const_500_0 ≤ 0 ∧ ___const_500_0 − ___const_500_0 ≤ 0
4	3	2:	− st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − s_17_0 + s_17_0 ≤ 0 ∧ s_17_0 − s_17_0 ≤ 0 ∧ − s_16_0 + s_16_0 ≤ 0 ∧ s_16_0 − s_16_0 ≤ 0 ∧ − rt_11_post + rt_11_post ≤ 0 ∧ rt_11_post − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_0 ≤ 0 ∧ rt_11_0 − rt_11_0 ≤ 0 ∧ − j_15_post + j_15_post ≤ 0 ∧ j_15_post − j_15_post ≤ 0 ∧ − j_15_0 + j_15_0 ≤ 0 ∧ j_15_0 − j_15_0 ≤ 0 ∧ − i_13_post + i_13_post ≤ 0 ∧ i_13_post − i_13_post ≤ 0 ∧ − i_13_1 + i_13_1 ≤ 0 ∧ i_13_1 − i_13_1 ≤ 0 ∧ − i_13_0 + i_13_0 ≤ 0 ∧ i_13_0 − i_13_0 ≤ 0 ∧ − ___const_500_0 + ___const_500_0 ≤ 0 ∧ ___const_500_0 − ___const_500_0 ≤ 0
1	4	3:	0 ≤ 0 ∧ 0 ≤ 0 ∧ ___const_500_0 − i_13_0 ≤ 0 ∧ rt_11_post − st_14_0 ≤ 0 ∧ − rt_11_post + st_14_0 ≤ 0 ∧ rt_11_0 − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_post ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − s_17_0 + s_17_0 ≤ 0 ∧ s_17_0 − s_17_0 ≤ 0 ∧ − s_16_0 + s_16_0 ≤ 0 ∧ s_16_0 − s_16_0 ≤ 0 ∧ − j_15_post + j_15_post ≤ 0 ∧ j_15_post − j_15_post ≤ 0 ∧ − j_15_0 + j_15_0 ≤ 0 ∧ j_15_0 − j_15_0 ≤ 0 ∧ − i_13_post + i_13_post ≤ 0 ∧ i_13_post − i_13_post ≤ 0 ∧ − i_13_1 + i_13_1 ≤ 0 ∧ i_13_1 − i_13_1 ≤ 0 ∧ − i_13_0 + i_13_0 ≤ 0 ∧ i_13_0 − i_13_0 ≤ 0 ∧ − ___const_500_0 + ___const_500_0 ≤ 0 ∧ ___const_500_0 − ___const_500_0 ≤ 0
1	5	2:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − ___const_500_0 + i_13_0 ≤ 0 ∧ −1 − i_13_0 + i_13_post ≤ 0 ∧ 1 + i_13_0 − i_13_post ≤ 0 ∧ i_13_0 − i_13_post ≤ 0 ∧ − i_13_0 + i_13_post ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − s_17_0 + s_17_0 ≤ 0 ∧ s_17_0 − s_17_0 ≤ 0 ∧ − s_16_0 + s_16_0 ≤ 0 ∧ s_16_0 − s_16_0 ≤ 0 ∧ − rt_11_post + rt_11_post ≤ 0 ∧ rt_11_post − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_0 ≤ 0 ∧ rt_11_0 − rt_11_0 ≤ 0 ∧ − j_15_post + j_15_post ≤ 0 ∧ j_15_post − j_15_post ≤ 0 ∧ − j_15_0 + j_15_0 ≤ 0 ∧ j_15_0 − j_15_0 ≤ 0 ∧ − i_13_1 + i_13_1 ≤ 0 ∧ i_13_1 − i_13_1 ≤ 0 ∧ − ___const_500_0 + ___const_500_0 ≤ 0 ∧ ___const_500_0 − ___const_500_0 ≤ 0
5	6	0:	− st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − s_17_0 + s_17_0 ≤ 0 ∧ s_17_0 − s_17_0 ≤ 0 ∧ − s_16_0 + s_16_0 ≤ 0 ∧ s_16_0 − s_16_0 ≤ 0 ∧ − rt_11_post + rt_11_post ≤ 0 ∧ rt_11_post − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_0 ≤ 0 ∧ rt_11_0 − rt_11_0 ≤ 0 ∧ − j_15_post + j_15_post ≤ 0 ∧ j_15_post − j_15_post ≤ 0 ∧ − j_15_0 + j_15_0 ≤ 0 ∧ j_15_0 − j_15_0 ≤ 0 ∧ − i_13_post + i_13_post ≤ 0 ∧ i_13_post − i_13_post ≤ 0 ∧ − i_13_1 + i_13_1 ≤ 0 ∧ i_13_1 − i_13_1 ≤ 0 ∧ − i_13_0 + i_13_0 ≤ 0 ∧ i_13_0 − i_13_0 ≤ 0 ∧ − ___const_500_0 + ___const_500_0 ≤ 0 ∧ ___const_500_0 − ___const_500_0 ≤ 0

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

− st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − s_17_0 + s_17_0 ≤ 0 ∧ s_17_0 − s_17_0 ≤ 0 ∧ − s_16_0 + s_16_0 ≤ 0 ∧ s_16_0 − s_16_0 ≤ 0 ∧ − rt_11_post + rt_11_post ≤ 0 ∧ rt_11_post − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_0 ≤ 0 ∧ rt_11_0 − rt_11_0 ≤ 0 ∧ − j_15_post + j_15_post ≤ 0 ∧ j_15_post − j_15_post ≤ 0 ∧ − j_15_0 + j_15_0 ≤ 0 ∧ j_15_0 − j_15_0 ≤ 0 ∧ − i_13_post + i_13_post ≤ 0 ∧ i_13_post − i_13_post ≤ 0 ∧ − i_13_1 + i_13_1 ≤ 0 ∧ i_13_1 − i_13_1 ≤ 0 ∧ − i_13_0 + i_13_0 ≤ 0 ∧ i_13_0 − i_13_0 ≤ 0 ∧ − ___const_500_0 + ___const_500_0 ≤ 0 ∧ ___const_500_0 − ___const_500_0 ≤ 0

and for every transition t, a duplicate t is considered.

2 Transition Removal

We remove transitions 0, 1, 4, 5, 6 using the following ranking functions, which are bounded by −15.

5:	0
0:	0
1:	0
2:	0
4:	0
3:	0
5:	−6
0:	−7
1:	−8
2:	−9
4:	−9
2_var_snapshot:	−9
2*:	−9
3:	−13

3 Location Addition

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

2* 10 2: − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − s_17_0 + s_17_0 ≤ 0 ∧ s_17_0 − s_17_0 ≤ 0 ∧ − s_16_0 + s_16_0 ≤ 0 ∧ s_16_0 − s_16_0 ≤ 0 ∧ − rt_11_post + rt_11_post ≤ 0 ∧ rt_11_post − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_0 ≤ 0 ∧ rt_11_0 − rt_11_0 ≤ 0 ∧ − j_15_post + j_15_post ≤ 0 ∧ j_15_post − j_15_post ≤ 0 ∧ − j_15_0 + j_15_0 ≤ 0 ∧ j_15_0 − j_15_0 ≤ 0 ∧ − i_13_post + i_13_post ≤ 0 ∧ i_13_post − i_13_post ≤ 0 ∧ − i_13_1 + i_13_1 ≤ 0 ∧ i_13_1 − i_13_1 ≤ 0 ∧ − i_13_0 + i_13_0 ≤ 0 ∧ i_13_0 − i_13_0 ≤ 0 ∧ − ___const_500_0 + ___const_500_0 ≤ 0 ∧ ___const_500_0 − ___const_500_0 ≤ 0

4 Location Addition

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

2 8 2_var_snapshot: − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − s_17_0 + s_17_0 ≤ 0 ∧ s_17_0 − s_17_0 ≤ 0 ∧ − s_16_0 + s_16_0 ≤ 0 ∧ s_16_0 − s_16_0 ≤ 0 ∧ − rt_11_post + rt_11_post ≤ 0 ∧ rt_11_post − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_0 ≤ 0 ∧ rt_11_0 − rt_11_0 ≤ 0 ∧ − j_15_post + j_15_post ≤ 0 ∧ j_15_post − j_15_post ≤ 0 ∧ − j_15_0 + j_15_0 ≤ 0 ∧ j_15_0 − j_15_0 ≤ 0 ∧ − i_13_post + i_13_post ≤ 0 ∧ i_13_post − i_13_post ≤ 0 ∧ − i_13_1 + i_13_1 ≤ 0 ∧ i_13_1 − i_13_1 ≤ 0 ∧ − i_13_0 + i_13_0 ≤ 0 ∧ i_13_0 − i_13_0 ≤ 0 ∧ − ___const_500_0 + ___const_500_0 ≤ 0 ∧ ___const_500_0 − ___const_500_0 ≤ 0

5 SCC Decomposition

We consider subproblems for each of the 1 SCC(s) of the program graph.

5.1 SCC Subproblem 1/1

Here we consider the SCC { 2, 4, 2_var_snapshot, 2* }.

5.1.1 Transition Removal

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

2:	4⋅___const_500_0 − 4⋅i_13_0
4:	2 + 4⋅___const_500_0 − 4⋅i_13_0
2_var_snapshot:	−1 + 4⋅___const_500_0 − 4⋅i_13_0
2*:	1 + 4⋅___const_500_0 − 4⋅i_13_0

5.1.2 Transition Removal

We remove transitions 8, 10, 3 using the following ranking functions, which are bounded by −1.

2:	0
4:	2
2_var_snapshot:	−1
2*:	1

5.1.3 Splitting Cut-Point Transitions

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

5.1.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 7.

5.1.3.1.1 Splitting Cut-Point Transitions

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

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