LTS Termination Proof

by T2Cert

Input

Integer Transition System

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
0 10 0: y_post + y_post ≤ 0y_posty_post ≤ 0y_0 + y_0 ≤ 0y_0y_0 ≤ 0x_post + x_post ≤ 0x_postx_post ≤ 0x_0 + x_0 ≤ 0x_0x_0 ≤ 0Result_post + Result_post ≤ 0Result_postResult_post ≤ 0Result_0 + Result_0 ≤ 0Result_0Result_0 ≤ 0
and for every transition t, a duplicate t is considered.

2 Transition Removal

We remove transitions 0, 8, 9 using the following ranking functions, which are bounded by −13.

7: 0
6: 0
0: 0
2: 0
3: 0
4: 0
5: 0
1: 0
7: −5
6: −6
0: −7
2: −7
3: −7
4: −7
5: −7
0_var_snapshot: −7
0*: −7
1: −11
Hints:
11 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
1 lexWeak[ [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] ]
3 lexWeak[ [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] ]
5 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
6 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
7 lexWeak[ [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] ]
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] ]
9 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] ]

3 Location Addition

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

0* 13 0: y_post + y_post ≤ 0y_posty_post ≤ 0y_0 + y_0 ≤ 0y_0y_0 ≤ 0x_post + x_post ≤ 0x_postx_post ≤ 0x_0 + x_0 ≤ 0x_0x_0 ≤ 0Result_post + Result_post ≤ 0Result_postResult_post ≤ 0Result_0 + Result_0 ≤ 0Result_0Result_0 ≤ 0

4 Location Addition

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

0 11 0_var_snapshot: y_post + y_post ≤ 0y_posty_post ≤ 0y_0 + y_0 ≤ 0y_0y_0 ≤ 0x_post + x_post ≤ 0x_postx_post ≤ 0x_0 + x_0 ≤ 0x_0x_0 ≤ 0Result_post + Result_post ≤ 0Result_postResult_post ≤ 0Result_0 + Result_0 ≤ 0Result_0Result_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 { 0, 2, 3, 4, 5, 0_var_snapshot, 0* }.

5.1.1 Transition Removal

We remove transitions 2, 6 using the following ranking functions, which are bounded by 2.

0: −2 − 6⋅y_0
2: −4 − 6⋅y_0
3: −5 − 6⋅y_0
4: −6⋅y_0
5: −6⋅y_0
0_var_snapshot: −3 − 6⋅y_0
0*: −1 − 6⋅y_0
Hints:
11 lexWeak[ [0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0] ]
13 lexWeak[ [0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0] ]
1 lexWeak[ [0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0] ]
2 lexStrict[ [0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0] , [6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
3 lexWeak[ [0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0] ]
4 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 6, 0, 0, 0, 0] ]
5 lexWeak[ [0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0] ]
6 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
7 lexWeak[ [0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0] ]

5.1.2 Transition Removal

We remove transition 1 using the following ranking functions, which are bounded by 2.

0: −2 − 5⋅x_0
2: −3 − 5⋅x_0
3: −4 − 5⋅x_0
4: −5⋅x_0
5: −5⋅x_0
0_var_snapshot: −2 − 5⋅x_0
0*: −1 − 5⋅x_0
Hints:
11 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0] ]
13 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0] ]
1 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0] , [5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
3 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0] ]
4 lexWeak[ [0, 0, 0, 0, 0, 5, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0] ]
5 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0] ]
7 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0] ]

5.1.3 Transition Removal

We remove transitions 11, 13, 3, 4, 5, 7 using the following ranking functions, which are bounded by −3.

0: −2
2: 2
3: 1
4: 0
5: 0
0_var_snapshot: −3
0*: −1
Hints:
11 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] ]
13 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] ]
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] ]
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] ]
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] ]
7 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] ]

5.1.4 Splitting Cut-Point Transitions

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

5.1.4.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 10.

5.1.4.1.1 Splitting Cut-Point Transitions

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

Tool configuration

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