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
3: 1 − t_22_post ≤ 01 − y_21_post ≤ 01 − t_22_0 ≤ 01 − y_21_0 ≤ 0
4: TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
1 6 1: y_21_post + y_21_post ≤ 0y_21_posty_21_post ≤ 0y_21_0 + y_21_0 ≤ 0y_21_0y_21_0 ≤ 0y_15_post + y_15_post ≤ 0y_15_posty_15_post ≤ 0y_15_1 + y_15_1 ≤ 0y_15_1y_15_1 ≤ 0y_15_0 + y_15_0 ≤ 0y_15_0y_15_0 ≤ 0x_13_post + x_13_post ≤ 0x_13_postx_13_post ≤ 0x_13_1 + x_13_1 ≤ 0x_13_1x_13_1 ≤ 0x_13_0 + x_13_0 ≤ 0x_13_0x_13_0 ≤ 0temp0_14_0 + temp0_14_0 ≤ 0temp0_14_0temp0_14_0 ≤ 0t_22_post + t_22_post ≤ 0t_22_postt_22_post ≤ 0t_22_0 + t_22_0 ≤ 0t_22_0t_22_0 ≤ 0t_16_post + t_16_post ≤ 0t_16_postt_16_post ≤ 0t_16_1 + t_16_1 ≤ 0t_16_1t_16_1 ≤ 0t_16_0 + t_16_0 ≤ 0t_16_0t_16_0 ≤ 0result_11_post + result_11_post ≤ 0result_11_postresult_11_post ≤ 0result_11_0 + result_11_0 ≤ 0result_11_0result_11_0 ≤ 0
and for every transition t, a duplicate t is considered.

3 Transition Removal

We remove transitions 0, 1, 2, 5 using the following ranking functions, which are bounded by −13.

4: 0
0: 0
1: 0
3: 0
2: 0
4: −5
0: −6
1: −7
3: −7
1_var_snapshot: −7
1*: −7
2: −11
Hints:
7 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] ]
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, 0, 0, 0, 0, 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, 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, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
2 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, 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, 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* 9 1: y_21_post + y_21_post ≤ 0y_21_posty_21_post ≤ 0y_21_0 + y_21_0 ≤ 0y_21_0y_21_0 ≤ 0y_15_post + y_15_post ≤ 0y_15_posty_15_post ≤ 0y_15_1 + y_15_1 ≤ 0y_15_1y_15_1 ≤ 0y_15_0 + y_15_0 ≤ 0y_15_0y_15_0 ≤ 0x_13_post + x_13_post ≤ 0x_13_postx_13_post ≤ 0x_13_1 + x_13_1 ≤ 0x_13_1x_13_1 ≤ 0x_13_0 + x_13_0 ≤ 0x_13_0x_13_0 ≤ 0temp0_14_0 + temp0_14_0 ≤ 0temp0_14_0temp0_14_0 ≤ 0t_22_post + t_22_post ≤ 0t_22_postt_22_post ≤ 0t_22_0 + t_22_0 ≤ 0t_22_0t_22_0 ≤ 0t_16_post + t_16_post ≤ 0t_16_postt_16_post ≤ 0t_16_1 + t_16_1 ≤ 0t_16_1t_16_1 ≤ 0t_16_0 + t_16_0 ≤ 0t_16_0t_16_0 ≤ 0result_11_post + result_11_post ≤ 0result_11_postresult_11_post ≤ 0result_11_0 + result_11_0 ≤ 0result_11_0result_11_0 ≤ 0

5 Location Addition

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

1 7 1_var_snapshot: y_21_post + y_21_post ≤ 0y_21_posty_21_post ≤ 0y_21_0 + y_21_0 ≤ 0y_21_0y_21_0 ≤ 0y_15_post + y_15_post ≤ 0y_15_posty_15_post ≤ 0y_15_1 + y_15_1 ≤ 0y_15_1y_15_1 ≤ 0y_15_0 + y_15_0 ≤ 0y_15_0y_15_0 ≤ 0x_13_post + x_13_post ≤ 0x_13_postx_13_post ≤ 0x_13_1 + x_13_1 ≤ 0x_13_1x_13_1 ≤ 0x_13_0 + x_13_0 ≤ 0x_13_0x_13_0 ≤ 0temp0_14_0 + temp0_14_0 ≤ 0temp0_14_0temp0_14_0 ≤ 0t_22_post + t_22_post ≤ 0t_22_postt_22_post ≤ 0t_22_0 + t_22_0 ≤ 0t_22_0t_22_0 ≤ 0t_16_post + t_16_post ≤ 0t_16_postt_16_post ≤ 0t_16_1 + t_16_1 ≤ 0t_16_1t_16_1 ≤ 0t_16_0 + t_16_0 ≤ 0t_16_0t_16_0 ≤ 0result_11_post + result_11_post ≤ 0result_11_postresult_11_post ≤ 0result_11_0 + result_11_0 ≤ 0result_11_0result_11_0 ≤ 0

6 SCC Decomposition

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

6.1 SCC Subproblem 1/1

Here we consider the SCC { 1, 3, 1_var_snapshot, 1* }.

6.1.1 Transition Removal

We remove transition 3 using the following ranking functions, which are bounded by 0.

1: −2 + x_13_0 + 3⋅y_15_0
3: x_13_0 + 3⋅y_15_0
1_var_snapshot: −3 + x_13_0 + 3⋅y_15_0
1*: −1 + x_13_0 + 3⋅y_15_0
Hints:
7 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
9 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 1, 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, 1, 0, 3, 0, 1, 0, 3, 0, 1, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 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, 0, 3, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 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, 3, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

6.1.2 Transition Removal

We remove transitions 7, 9, 4 using the following ranking functions, which are bounded by −2.

1: −1
3: y_21_0
1_var_snapshot: −2
1*: 0
Hints:
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, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 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, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

6.1.3 Splitting Cut-Point Transitions

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

6.1.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 6.

6.1.3.1.1 Splitting Cut-Point Transitions

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

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