LTS Termination Proof

by T2Cert

Input

Integer Transition System

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
1 7 1: st_14_0 + st_14_0 ≤ 0st_14_0st_14_0 ≤ 0rt_11_post + rt_11_post ≤ 0rt_11_postrt_11_post ≤ 0rt_11_0 + rt_11_0 ≤ 0rt_11_0rt_11_0 ≤ 0i_21_post + i_21_post ≤ 0i_21_posti_21_post ≤ 0i_21_0 + i_21_0 ≤ 0i_21_0i_21_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 ≤ 0a_20_post + a_20_post ≤ 0a_20_posta_20_post ≤ 0a_20_0 + a_20_0 ≤ 0a_20_0a_20_0 ≤ 0___const_10_0 + ___const_10_0 ≤ 0___const_10_0___const_10_0 ≤ 0
and for every transition t, a duplicate t is considered.

2 Transition Removal

We remove transitions 0, 3, 6 using the following ranking functions, which are bounded by −13.

6: 0
0: 0
1: 0
5: 0
4: 0
6: −5
0: −6
1: −7
5: −7
1_var_snapshot: −7
1*: −7
4: −11
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] ]
4 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, 0, 0, 0] ]
5 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] ]
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, 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.

1* 10 1: st_14_0 + st_14_0 ≤ 0st_14_0st_14_0 ≤ 0rt_11_post + rt_11_post ≤ 0rt_11_postrt_11_post ≤ 0rt_11_0 + rt_11_0 ≤ 0rt_11_0rt_11_0 ≤ 0i_21_post + i_21_post ≤ 0i_21_posti_21_post ≤ 0i_21_0 + i_21_0 ≤ 0i_21_0i_21_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 ≤ 0a_20_post + a_20_post ≤ 0a_20_posta_20_post ≤ 0a_20_0 + a_20_0 ≤ 0a_20_0a_20_0 ≤ 0___const_10_0 + ___const_10_0 ≤ 0___const_10_0___const_10_0 ≤ 0

4 Location Addition

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

1 8 1_var_snapshot: st_14_0 + st_14_0 ≤ 0st_14_0st_14_0 ≤ 0rt_11_post + rt_11_post ≤ 0rt_11_postrt_11_post ≤ 0rt_11_0 + rt_11_0 ≤ 0rt_11_0rt_11_0 ≤ 0i_21_post + i_21_post ≤ 0i_21_posti_21_post ≤ 0i_21_0 + i_21_0 ≤ 0i_21_0i_21_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 ≤ 0a_20_post + a_20_post ≤ 0a_20_posta_20_post ≤ 0a_20_0 + a_20_0 ≤ 0a_20_0a_20_0 ≤ 0___const_10_0 + ___const_10_0 ≤ 0___const_10_0___const_10_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 { 1, 5, 1_var_snapshot, 1* }.

5.1.1 Transition Removal

We remove transition 4 using the following ranking functions, which are bounded by 0.

1: −2 + 3⋅___const_10_0 − 3⋅i_13_0
5: 3⋅___const_10_0 − 3⋅i_13_0
1_var_snapshot: −2 + 3⋅___const_10_0 − 3⋅i_13_0
1*: −1 + 3⋅___const_10_0 − 3⋅i_13_0
Hints:
8 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 3, 0] ]
10 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 3, 0] ]
4 lexStrict[ [0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0] , [0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
5 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 3, 0] ]

5.1.2 Transition Removal

We remove transitions 8, 10, 5 using the following ranking functions, which are bounded by −1.

1: 0
5: 2
1_var_snapshot: −1
1*: 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] ]
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] ]

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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