LTS Termination Proof

by T2Cert

Input

Integer Transition System

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
3 17 3: x1_post + x1_post ≤ 0x1_postx1_post ≤ 0x1_0 + x1_0 ≤ 0x1_0x1_0 ≤ 0x0_post + x0_post ≤ 0x0_postx0_post ≤ 0x0_0 + x0_0 ≤ 0x0_0x0_0 ≤ 0oldX3_post + oldX3_post ≤ 0oldX3_postoldX3_post ≤ 0oldX3_0 + oldX3_0 ≤ 0oldX3_0oldX3_0 ≤ 0oldX2_post + oldX2_post ≤ 0oldX2_postoldX2_post ≤ 0oldX2_0 + oldX2_0 ≤ 0oldX2_0oldX2_0 ≤ 0oldX1_post + oldX1_post ≤ 0oldX1_postoldX1_post ≤ 0oldX1_0 + oldX1_0 ≤ 0oldX1_0oldX1_0 ≤ 0oldX0_post + oldX0_post ≤ 0oldX0_postoldX0_post ≤ 0oldX0_0 + oldX0_0 ≤ 0oldX0_0oldX0_0 ≤ 0___const_99_0 + ___const_99_0 ≤ 0___const_99_0___const_99_0 ≤ 0___const_5_0 + ___const_5_0 ≤ 0___const_5_0___const_5_0 ≤ 0___const_20_0 + ___const_20_0 ≤ 0___const_20_0___const_20_0 ≤ 0
5 24 5: x1_post + x1_post ≤ 0x1_postx1_post ≤ 0x1_0 + x1_0 ≤ 0x1_0x1_0 ≤ 0x0_post + x0_post ≤ 0x0_postx0_post ≤ 0x0_0 + x0_0 ≤ 0x0_0x0_0 ≤ 0oldX3_post + oldX3_post ≤ 0oldX3_postoldX3_post ≤ 0oldX3_0 + oldX3_0 ≤ 0oldX3_0oldX3_0 ≤ 0oldX2_post + oldX2_post ≤ 0oldX2_postoldX2_post ≤ 0oldX2_0 + oldX2_0 ≤ 0oldX2_0oldX2_0 ≤ 0oldX1_post + oldX1_post ≤ 0oldX1_postoldX1_post ≤ 0oldX1_0 + oldX1_0 ≤ 0oldX1_0oldX1_0 ≤ 0oldX0_post + oldX0_post ≤ 0oldX0_postoldX0_post ≤ 0oldX0_0 + oldX0_0 ≤ 0oldX0_0oldX0_0 ≤ 0___const_99_0 + ___const_99_0 ≤ 0___const_99_0___const_99_0 ≤ 0___const_5_0 + ___const_5_0 ≤ 0___const_5_0___const_5_0 ≤ 0___const_20_0 + ___const_20_0 ≤ 0___const_20_0___const_20_0 ≤ 0
and for every transition t, a duplicate t is considered.

2 Transition Removal

We remove transitions 0, 2, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 using the following ranking functions, which are bounded by −21.

8: 0
7: 0
6: 0
4: 0
5: 0
2: 0
3: 0
0: 0
1: 0
8: −8
7: −9
6: −10
4: −11
5: −11
5_var_snapshot: −11
5*: −11
2: −14
3: −14
3_var_snapshot: −14
3*: −14
0: −18
1: −19

3 Location Addition

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

3* 20 3: x1_post + x1_post ≤ 0x1_postx1_post ≤ 0x1_0 + x1_0 ≤ 0x1_0x1_0 ≤ 0x0_post + x0_post ≤ 0x0_postx0_post ≤ 0x0_0 + x0_0 ≤ 0x0_0x0_0 ≤ 0oldX3_post + oldX3_post ≤ 0oldX3_postoldX3_post ≤ 0oldX3_0 + oldX3_0 ≤ 0oldX3_0oldX3_0 ≤ 0oldX2_post + oldX2_post ≤ 0oldX2_postoldX2_post ≤ 0oldX2_0 + oldX2_0 ≤ 0oldX2_0oldX2_0 ≤ 0oldX1_post + oldX1_post ≤ 0oldX1_postoldX1_post ≤ 0oldX1_0 + oldX1_0 ≤ 0oldX1_0oldX1_0 ≤ 0oldX0_post + oldX0_post ≤ 0oldX0_postoldX0_post ≤ 0oldX0_0 + oldX0_0 ≤ 0oldX0_0oldX0_0 ≤ 0___const_99_0 + ___const_99_0 ≤ 0___const_99_0___const_99_0 ≤ 0___const_5_0 + ___const_5_0 ≤ 0___const_5_0___const_5_0 ≤ 0___const_20_0 + ___const_20_0 ≤ 0___const_20_0___const_20_0 ≤ 0

4 Location Addition

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

3 18 3_var_snapshot: x1_post + x1_post ≤ 0x1_postx1_post ≤ 0x1_0 + x1_0 ≤ 0x1_0x1_0 ≤ 0x0_post + x0_post ≤ 0x0_postx0_post ≤ 0x0_0 + x0_0 ≤ 0x0_0x0_0 ≤ 0oldX3_post + oldX3_post ≤ 0oldX3_postoldX3_post ≤ 0oldX3_0 + oldX3_0 ≤ 0oldX3_0oldX3_0 ≤ 0oldX2_post + oldX2_post ≤ 0oldX2_postoldX2_post ≤ 0oldX2_0 + oldX2_0 ≤ 0oldX2_0oldX2_0 ≤ 0oldX1_post + oldX1_post ≤ 0oldX1_postoldX1_post ≤ 0oldX1_0 + oldX1_0 ≤ 0oldX1_0oldX1_0 ≤ 0oldX0_post + oldX0_post ≤ 0oldX0_postoldX0_post ≤ 0oldX0_0 + oldX0_0 ≤ 0oldX0_0oldX0_0 ≤ 0___const_99_0 + ___const_99_0 ≤ 0___const_99_0___const_99_0 ≤ 0___const_5_0 + ___const_5_0 ≤ 0___const_5_0___const_5_0 ≤ 0___const_20_0 + ___const_20_0 ≤ 0___const_20_0___const_20_0 ≤ 0

5 Location Addition

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

5* 27 5: x1_post + x1_post ≤ 0x1_postx1_post ≤ 0x1_0 + x1_0 ≤ 0x1_0x1_0 ≤ 0x0_post + x0_post ≤ 0x0_postx0_post ≤ 0x0_0 + x0_0 ≤ 0x0_0x0_0 ≤ 0oldX3_post + oldX3_post ≤ 0oldX3_postoldX3_post ≤ 0oldX3_0 + oldX3_0 ≤ 0oldX3_0oldX3_0 ≤ 0oldX2_post + oldX2_post ≤ 0oldX2_postoldX2_post ≤ 0oldX2_0 + oldX2_0 ≤ 0oldX2_0oldX2_0 ≤ 0oldX1_post + oldX1_post ≤ 0oldX1_postoldX1_post ≤ 0oldX1_0 + oldX1_0 ≤ 0oldX1_0oldX1_0 ≤ 0oldX0_post + oldX0_post ≤ 0oldX0_postoldX0_post ≤ 0oldX0_0 + oldX0_0 ≤ 0oldX0_0oldX0_0 ≤ 0___const_99_0 + ___const_99_0 ≤ 0___const_99_0___const_99_0 ≤ 0___const_5_0 + ___const_5_0 ≤ 0___const_5_0___const_5_0 ≤ 0___const_20_0 + ___const_20_0 ≤ 0___const_20_0___const_20_0 ≤ 0

6 Location Addition

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

5 25 5_var_snapshot: x1_post + x1_post ≤ 0x1_postx1_post ≤ 0x1_0 + x1_0 ≤ 0x1_0x1_0 ≤ 0x0_post + x0_post ≤ 0x0_postx0_post ≤ 0x0_0 + x0_0 ≤ 0x0_0x0_0 ≤ 0oldX3_post + oldX3_post ≤ 0oldX3_postoldX3_post ≤ 0oldX3_0 + oldX3_0 ≤ 0oldX3_0oldX3_0 ≤ 0oldX2_post + oldX2_post ≤ 0oldX2_postoldX2_post ≤ 0oldX2_0 + oldX2_0 ≤ 0oldX2_0oldX2_0 ≤ 0oldX1_post + oldX1_post ≤ 0oldX1_postoldX1_post ≤ 0oldX1_0 + oldX1_0 ≤ 0oldX1_0oldX1_0 ≤ 0oldX0_post + oldX0_post ≤ 0oldX0_postoldX0_post ≤ 0oldX0_0 + oldX0_0 ≤ 0oldX0_0oldX0_0 ≤ 0___const_99_0 + ___const_99_0 ≤ 0___const_99_0___const_99_0 ≤ 0___const_5_0 + ___const_5_0 ≤ 0___const_5_0___const_5_0 ≤ 0___const_20_0 + ___const_20_0 ≤ 0___const_20_0___const_20_0 ≤ 0

7 SCC Decomposition

We consider subproblems for each of the 2 SCC(s) of the program graph.

7.1 SCC Subproblem 1/2

Here we consider the SCC { 2, 3, 3_var_snapshot, 3* }.

7.1.1 Transition Removal

We remove transition 3 using the following ranking functions, which are bounded by −1.

2: −3 + 4⋅___const_20_0 − 4⋅x1_0
3: 3 + 4⋅___const_20_0 − 4⋅x1_0
3_var_snapshot: 4⋅___const_20_0 − 4⋅x1_0
3*: 6 + 4⋅___const_20_0 − 4⋅x1_0

7.1.2 Transition Removal

We remove transitions 18, 20, 1 using the following ranking functions, which are bounded by −1.

2: 2
3: 0
3_var_snapshot: −1
3*: 1

7.1.3 Splitting Cut-Point Transitions

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

7.1.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 17.

7.1.3.1.1 Splitting Cut-Point Transitions

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

7.2 SCC Subproblem 2/2

Here we consider the SCC { 4, 5, 5_var_snapshot, 5* }.

7.2.1 Transition Removal

We remove transition 6 using the following ranking functions, which are bounded by −1.

4: −1 + 4⋅___const_99_0 − 4⋅x0_0
5: 1 + 4⋅___const_99_0 − 4⋅x0_0
5_var_snapshot: 4⋅___const_99_0 − 4⋅x0_0
5*: 2 + 4⋅___const_99_0 − 4⋅x0_0

7.2.2 Transition Removal

We remove transitions 25, 27, 4 using the following ranking functions, which are bounded by −2.

4: 1
5: −1
5_var_snapshot: −2
5*: 0

7.2.3 Splitting Cut-Point Transitions

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

7.2.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 24.

7.2.3.1.1 Splitting Cut-Point Transitions

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

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