LTS Termination Proof

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Input

Integer Transition System

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
2 7 2: st_14_0 + st_14_0 ≤ 0st_14_0st_14_0 ≤ 0s_17_0 + s_17_0 ≤ 0s_17_0s_17_0 ≤ 0s_16_0 + s_16_0 ≤ 0s_16_0s_16_0 ≤ 0rt_11_post + rt_11_post ≤ 0rt_11_postrt_11_post ≤ 0rt_11_0 + rt_11_0 ≤ 0rt_11_0rt_11_0 ≤ 0j_15_post + j_15_post ≤ 0j_15_postj_15_post ≤ 0j_15_0 + j_15_0 ≤ 0j_15_0j_15_0 ≤ 0i_13_post + i_13_post ≤ 0i_13_posti_13_post ≤ 0i_13_1 + i_13_1 ≤ 0i_13_1i_13_1 ≤ 0i_13_0 + i_13_0 ≤ 0i_13_0i_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
Hints:
8 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
2 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
3 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
0 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
1 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
4 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
5 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
6 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

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 ≤ 0st_14_0st_14_0 ≤ 0s_17_0 + s_17_0 ≤ 0s_17_0s_17_0 ≤ 0s_16_0 + s_16_0 ≤ 0s_16_0s_16_0 ≤ 0rt_11_post + rt_11_post ≤ 0rt_11_postrt_11_post ≤ 0rt_11_0 + rt_11_0 ≤ 0rt_11_0rt_11_0 ≤ 0j_15_post + j_15_post ≤ 0j_15_postj_15_post ≤ 0j_15_0 + j_15_0 ≤ 0j_15_0j_15_0 ≤ 0i_13_post + i_13_post ≤ 0i_13_posti_13_post ≤ 0i_13_1 + i_13_1 ≤ 0i_13_1i_13_1 ≤ 0i_13_0 + i_13_0 ≤ 0i_13_0i_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 ≤ 0st_14_0st_14_0 ≤ 0s_17_0 + s_17_0 ≤ 0s_17_0s_17_0 ≤ 0s_16_0 + s_16_0 ≤ 0s_16_0s_16_0 ≤ 0rt_11_post + rt_11_post ≤ 0rt_11_postrt_11_post ≤ 0rt_11_0 + rt_11_0 ≤ 0rt_11_0rt_11_0 ≤ 0j_15_post + j_15_post ≤ 0j_15_postj_15_post ≤ 0j_15_0 + j_15_0 ≤ 0j_15_0j_15_0 ≤ 0i_13_post + i_13_post ≤ 0i_13_posti_13_post ≤ 0i_13_1 + i_13_1 ≤ 0i_13_1i_13_1 ≤ 0i_13_0 + i_13_0 ≤ 0i_13_0i_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
Hints:
8 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 4, 0] ]
10 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 4, 0] ]
2 lexStrict[ [0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0] , [0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
3 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 4, 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
Hints:
8 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
10 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
3 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

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.

Tool configuration

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