LTS Termination Proof

by T2Cert

Input

Integer Transition System

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
1 17 1: 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
3 24 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
and for every transition t, a duplicate t is considered.

2 Transition Removal

We remove transitions 2, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 using the following ranking functions, which are bounded by −19.

8: 0
7: 0
6: 0
0: 0
1: 0
2: 0
3: 0
4: 0
5: 0
8: −7
7: −8
6: −9
0: −10
1: −10
2: −10
3: −10
1_var_snapshot: −10
1*: −10
3_var_snapshot: −10
3*: −10
4: −16
5: −17
Hints:
18 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] ]
25 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 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] ]
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, 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] ]
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] ]
6 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] ]
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] ]
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, 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, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 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, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 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] ]
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] ]
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] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
12 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] ]
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] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
14 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] ]
16 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 Location Addition

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

1* 20 1: 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

4 Location Addition

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

1 18 1_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

5 Location Addition

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

3* 27 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

6 Location Addition

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

3 25 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

7 SCC Decomposition

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

7.1 SCC Subproblem 1/1

Here we consider the SCC { 0, 1, 2, 3, 1_var_snapshot, 1*, 3_var_snapshot, 3* }.

7.1.1 Transition Removal

We remove transition 6 using the following ranking functions, which are bounded by −82.

0: −6 − 9⋅x0_0
1: 1 − 9⋅x0_0
2: −4 − 9⋅x0_0
3: −4 − 9⋅x0_0
1_var_snapshot: −9⋅x0_0
1*: 2 − 9⋅x0_0
3_var_snapshot: −4 − 9⋅x0_0
3*: −4 − 9⋅x0_0
Hints:
18 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
20 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
25 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
27 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
0 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0] ]
1 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
3 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
4 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
6 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

7.1.2 Transition Removal

We remove transition 4 using the following ranking functions, which are bounded by −34.

0: 35 − 6⋅x1_0
1: 33 − 3⋅oldX1_0 − 3⋅oldX1_post
2: −1 − 3⋅x1_0
3: −3⋅x1_0
1_var_snapshot: 32 − 3⋅oldX1_0 − 3⋅oldX1_post
1*: 34 − 3⋅oldX1_0 − 3⋅oldX1_post
3_var_snapshot: −3⋅x1_0
3*: 1 − 3⋅x1_0
Hints:
18 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 3, 0, 0, 0, 0] ]
20 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 3, 0, 0, 0, 0] ]
25 lexWeak[ [0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
27 lexWeak[ [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 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
1 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0] ]
3 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 3, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0] ]
4 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 3, 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, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

7.1.3 Transition Removal

We remove transitions 18, 20, 25, 27, 0, 1, 3 using the following ranking functions, which are bounded by −7.

0: −4
1: −6
2: 0
3: −2
1_var_snapshot: −7
1*: −5
3_var_snapshot: −3
3*: −1
Hints:
18 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] ]
20 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] ]
25 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] ]
27 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 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] ]
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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

7.1.4 Splitting Cut-Point Transitions

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

7.1.4.1 Cut-Point Subproblem 1/2

Here we consider cut-point transition 17.

7.1.4.1.1 Splitting Cut-Point Transitions

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

7.1.4.2 Cut-Point Subproblem 2/2

Here we consider cut-point transition 24.

7.1.4.2.1 Splitting Cut-Point Transitions

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

Tool configuration

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