LTS Termination Proof

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Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

0: TRUE
1: TRUE
2: TRUE
4: 1 − y_33_post ≤ 01 − y_33_0 ≤ 0
5: TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
1 7 1: y_33_post + y_33_post ≤ 0y_33_posty_33_post ≤ 0y_33_0 + y_33_0 ≤ 0y_33_0y_33_0 ≤ 0y_28_post + y_28_post ≤ 0y_28_posty_28_post ≤ 0y_28_0 + y_28_0 ≤ 0y_28_0y_28_0 ≤ 0y_16_post + y_16_post ≤ 0y_16_posty_16_post ≤ 0y_16_0 + y_16_0 ≤ 0y_16_0y_16_0 ≤ 0x_32_post + x_32_post ≤ 0x_32_postx_32_post ≤ 0x_32_0 + x_32_0 ≤ 0x_32_0x_32_0 ≤ 0x_27_post + x_27_post ≤ 0x_27_postx_27_post ≤ 0x_27_0 + x_27_0 ≤ 0x_27_0x_27_0 ≤ 0x_13_post + x_13_post ≤ 0x_13_postx_13_post ≤ 0x_13_0 + x_13_0 ≤ 0x_13_0x_13_0 ≤ 0
2 14 2: y_33_post + y_33_post ≤ 0y_33_posty_33_post ≤ 0y_33_0 + y_33_0 ≤ 0y_33_0y_33_0 ≤ 0y_28_post + y_28_post ≤ 0y_28_posty_28_post ≤ 0y_28_0 + y_28_0 ≤ 0y_28_0y_28_0 ≤ 0y_16_post + y_16_post ≤ 0y_16_posty_16_post ≤ 0y_16_0 + y_16_0 ≤ 0y_16_0y_16_0 ≤ 0x_32_post + x_32_post ≤ 0x_32_postx_32_post ≤ 0x_32_0 + x_32_0 ≤ 0x_32_0x_32_0 ≤ 0x_27_post + x_27_post ≤ 0x_27_postx_27_post ≤ 0x_27_0 + x_27_0 ≤ 0x_27_0x_27_0 ≤ 0x_13_post + x_13_post ≤ 0x_13_postx_13_post ≤ 0x_13_0 + x_13_0 ≤ 0x_13_0x_13_0 ≤ 0
and for every transition t, a duplicate t is considered.

3 Transition Removal

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

5: 0
0: 0
1: 0
2: 0
4: 0
5: −4
0: −5
1: −6
2: −6
4: −6
1_var_snapshot: −6
1*: −6
2_var_snapshot: −6
2*: −6
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, 0, 0] ]
15 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] ]
1 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, 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, 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, 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, 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] ]
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, 0, 0, 0, 0] ]

4 Location Addition

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

1* 10 1: y_33_post + y_33_post ≤ 0y_33_posty_33_post ≤ 0y_33_0 + y_33_0 ≤ 0y_33_0y_33_0 ≤ 0y_28_post + y_28_post ≤ 0y_28_posty_28_post ≤ 0y_28_0 + y_28_0 ≤ 0y_28_0y_28_0 ≤ 0y_16_post + y_16_post ≤ 0y_16_posty_16_post ≤ 0y_16_0 + y_16_0 ≤ 0y_16_0y_16_0 ≤ 0x_32_post + x_32_post ≤ 0x_32_postx_32_post ≤ 0x_32_0 + x_32_0 ≤ 0x_32_0x_32_0 ≤ 0x_27_post + x_27_post ≤ 0x_27_postx_27_post ≤ 0x_27_0 + x_27_0 ≤ 0x_27_0x_27_0 ≤ 0x_13_post + x_13_post ≤ 0x_13_postx_13_post ≤ 0x_13_0 + x_13_0 ≤ 0x_13_0x_13_0 ≤ 0

5 Location Addition

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

1 8 1_var_snapshot: y_33_post + y_33_post ≤ 0y_33_posty_33_post ≤ 0y_33_0 + y_33_0 ≤ 0y_33_0y_33_0 ≤ 0y_28_post + y_28_post ≤ 0y_28_posty_28_post ≤ 0y_28_0 + y_28_0 ≤ 0y_28_0y_28_0 ≤ 0y_16_post + y_16_post ≤ 0y_16_posty_16_post ≤ 0y_16_0 + y_16_0 ≤ 0y_16_0y_16_0 ≤ 0x_32_post + x_32_post ≤ 0x_32_postx_32_post ≤ 0x_32_0 + x_32_0 ≤ 0x_32_0x_32_0 ≤ 0x_27_post + x_27_post ≤ 0x_27_postx_27_post ≤ 0x_27_0 + x_27_0 ≤ 0x_27_0x_27_0 ≤ 0x_13_post + x_13_post ≤ 0x_13_postx_13_post ≤ 0x_13_0 + x_13_0 ≤ 0x_13_0x_13_0 ≤ 0

6 Location Addition

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

2* 17 2: y_33_post + y_33_post ≤ 0y_33_posty_33_post ≤ 0y_33_0 + y_33_0 ≤ 0y_33_0y_33_0 ≤ 0y_28_post + y_28_post ≤ 0y_28_posty_28_post ≤ 0y_28_0 + y_28_0 ≤ 0y_28_0y_28_0 ≤ 0y_16_post + y_16_post ≤ 0y_16_posty_16_post ≤ 0y_16_0 + y_16_0 ≤ 0y_16_0y_16_0 ≤ 0x_32_post + x_32_post ≤ 0x_32_postx_32_post ≤ 0x_32_0 + x_32_0 ≤ 0x_32_0x_32_0 ≤ 0x_27_post + x_27_post ≤ 0x_27_postx_27_post ≤ 0x_27_0 + x_27_0 ≤ 0x_27_0x_27_0 ≤ 0x_13_post + x_13_post ≤ 0x_13_postx_13_post ≤ 0x_13_0 + x_13_0 ≤ 0x_13_0x_13_0 ≤ 0

7 Location Addition

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

2 15 2_var_snapshot: y_33_post + y_33_post ≤ 0y_33_posty_33_post ≤ 0y_33_0 + y_33_0 ≤ 0y_33_0y_33_0 ≤ 0y_28_post + y_28_post ≤ 0y_28_posty_28_post ≤ 0y_28_0 + y_28_0 ≤ 0y_28_0y_28_0 ≤ 0y_16_post + y_16_post ≤ 0y_16_posty_16_post ≤ 0y_16_0 + y_16_0 ≤ 0y_16_0y_16_0 ≤ 0x_32_post + x_32_post ≤ 0x_32_postx_32_post ≤ 0x_32_0 + x_32_0 ≤ 0x_32_0x_32_0 ≤ 0x_27_post + x_27_post ≤ 0x_27_postx_27_post ≤ 0x_27_0 + x_27_0 ≤ 0x_27_0x_27_0 ≤ 0x_13_post + x_13_post ≤ 0x_13_postx_13_post ≤ 0x_13_0 + x_13_0 ≤ 0x_13_0x_13_0 ≤ 0

8 SCC Decomposition

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

8.1 SCC Subproblem 1/1

Here we consider the SCC { 1, 2, 4, 1_var_snapshot, 1*, 2_var_snapshot, 2* }.

8.1.1 Transition Removal

We remove transition 1 using the following ranking functions, which are bounded by −9995.

1: −20000 + 10006⋅x_13_0
2: −5000 + 10006⋅x_13_0 − 5⋅y_16_0
4: 1 + 10006⋅x_13_0 − 5⋅y_16_0
1_var_snapshot: −20000 + 10006⋅x_13_0
1*: −15000 + 10006⋅x_13_0
2_var_snapshot: −10000 + 10006⋅x_13_0 − 5⋅y_16_0
2*: 10006⋅x_13_0 − 5⋅y_16_0
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, 10006, 0] ]
10 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10006, 0] ]
15 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10006, 0] ]
17 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10006, 0] ]
1 lexStrict[ [0, 0, 0, 0, 4, 1, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10006, 0] , [0, 0, 0, 0, 0, 0, 10006, 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, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10006, 0] ]
4 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 10006, 0, 0, 5, 0, 0, 0, 0, 0, 10006, 0, 0, 0, 0, 5, 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, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10006, 0] ]

8.1.2 Transition Removal

We remove transition 4 using the following ranking functions, which are bounded by 1.

1: −4 + 4⋅y_16_0
2: −1 + 4⋅y_16_0
4: 1 + 4⋅y_16_0
1_var_snapshot: −5 + 4⋅y_16_0
1*: −3 + 4⋅y_16_0
2_var_snapshot: −2 + 4⋅y_16_0
2*: 4⋅y_16_0
Hints:
8 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
10 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
15 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
17 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 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, 4, 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, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 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, 0, 0, 0] ]
5 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

8.1.3 Transition Removal

We remove transitions 8, 10, 15, 17, 3, 5 using the following ranking functions, which are bounded by −3.

1: −2
2: 1
4: 3⋅y_33_0
1_var_snapshot: −3
1*: −1
2_var_snapshot: 0
2*: 2
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, 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, 0, 0, 0, 0] ]
15 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] ]
17 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] ]
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, 0, 0] ]
5 lexStrict[ [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, 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, 0] ]

8.1.4 Splitting Cut-Point Transitions

We consider 2 subproblems corresponding to sets of cut-point transitions as follows.

8.1.4.1 Cut-Point Subproblem 1/2

Here we consider cut-point transition 7.

8.1.4.1.1 Splitting Cut-Point Transitions

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

8.1.4.2 Cut-Point Subproblem 2/2

Here we consider cut-point transition 14.

8.1.4.2.1 Splitting Cut-Point Transitions

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

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