LTS Termination Proof

by T2Cert

Input

Integer Transition System

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
2 6 2: y_13_post + y_13_post ≤ 0y_13_posty_13_post ≤ 0y_13_0 + y_13_0 ≤ 0y_13_0y_13_0 ≤ 0x_15_post + x_15_post ≤ 0x_15_postx_15_post ≤ 0x_15_0 + x_15_0 ≤ 0x_15_0x_15_0 ≤ 0st_14_0 + st_14_0 ≤ 0st_14_0st_14_0 ≤ 0rt_11_post + rt_11_post ≤ 0rt_11_postrt_11_post ≤ 0rt_11_0 + rt_11_0 ≤ 0rt_11_0rt_11_0 ≤ 0
and for every transition t, a duplicate t is considered.

2 Transition Removal

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

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

3 Location Addition

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

2* 9 2: y_13_post + y_13_post ≤ 0y_13_posty_13_post ≤ 0y_13_0 + y_13_0 ≤ 0y_13_0y_13_0 ≤ 0x_15_post + x_15_post ≤ 0x_15_postx_15_post ≤ 0x_15_0 + x_15_0 ≤ 0x_15_0x_15_0 ≤ 0st_14_0 + st_14_0 ≤ 0st_14_0st_14_0 ≤ 0rt_11_post + rt_11_post ≤ 0rt_11_postrt_11_post ≤ 0rt_11_0 + rt_11_0 ≤ 0rt_11_0rt_11_0 ≤ 0

4 Location Addition

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

2 7 2_var_snapshot: y_13_post + y_13_post ≤ 0y_13_posty_13_post ≤ 0y_13_0 + y_13_0 ≤ 0y_13_0y_13_0 ≤ 0x_15_post + x_15_post ≤ 0x_15_postx_15_post ≤ 0x_15_0 + x_15_0 ≤ 0x_15_0x_15_0 ≤ 0st_14_0 + st_14_0 ≤ 0st_14_0st_14_0 ≤ 0rt_11_post + rt_11_post ≤ 0rt_11_postrt_11_post ≤ 0rt_11_0 + rt_11_0 ≤ 0rt_11_0rt_11_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 { 2, 3, 2_var_snapshot, 2* }.

5.1.1 Splitting Cut-Point Transitions

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

5.1.1.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 6.

5.1.1.1.1 Fresh Variable Addition

The new variable __snapshot_2_y_13_post is introduced. The transition formulas are extended as follows:

7: __snapshot_2_y_13_posty_13_posty_13_post__snapshot_2_y_13_post
9: __snapshot_2_y_13_post__snapshot_2_y_13_post__snapshot_2_y_13_post__snapshot_2_y_13_post
3: __snapshot_2_y_13_post__snapshot_2_y_13_post__snapshot_2_y_13_post__snapshot_2_y_13_post
4: __snapshot_2_y_13_post__snapshot_2_y_13_post__snapshot_2_y_13_post__snapshot_2_y_13_post

5.1.1.1.2 Fresh Variable Addition

The new variable __snapshot_2_y_13_0 is introduced. The transition formulas are extended as follows:

7: __snapshot_2_y_13_0y_13_0y_13_0__snapshot_2_y_13_0
9: __snapshot_2_y_13_0__snapshot_2_y_13_0__snapshot_2_y_13_0__snapshot_2_y_13_0
3: __snapshot_2_y_13_0__snapshot_2_y_13_0__snapshot_2_y_13_0__snapshot_2_y_13_0
4: __snapshot_2_y_13_0__snapshot_2_y_13_0__snapshot_2_y_13_0__snapshot_2_y_13_0

5.1.1.1.3 Fresh Variable Addition

The new variable __snapshot_2_x_15_post is introduced. The transition formulas are extended as follows:

7: __snapshot_2_x_15_postx_15_postx_15_post__snapshot_2_x_15_post
9: __snapshot_2_x_15_post__snapshot_2_x_15_post__snapshot_2_x_15_post__snapshot_2_x_15_post
3: __snapshot_2_x_15_post__snapshot_2_x_15_post__snapshot_2_x_15_post__snapshot_2_x_15_post
4: __snapshot_2_x_15_post__snapshot_2_x_15_post__snapshot_2_x_15_post__snapshot_2_x_15_post

5.1.1.1.4 Fresh Variable Addition

The new variable __snapshot_2_x_15_0 is introduced. The transition formulas are extended as follows:

7: __snapshot_2_x_15_0x_15_0x_15_0__snapshot_2_x_15_0
9: __snapshot_2_x_15_0__snapshot_2_x_15_0__snapshot_2_x_15_0__snapshot_2_x_15_0
3: __snapshot_2_x_15_0__snapshot_2_x_15_0__snapshot_2_x_15_0__snapshot_2_x_15_0
4: __snapshot_2_x_15_0__snapshot_2_x_15_0__snapshot_2_x_15_0__snapshot_2_x_15_0

5.1.1.1.5 Fresh Variable Addition

The new variable __snapshot_2_st_14_0 is introduced. The transition formulas are extended as follows:

7: __snapshot_2_st_14_0st_14_0st_14_0__snapshot_2_st_14_0
9: __snapshot_2_st_14_0__snapshot_2_st_14_0__snapshot_2_st_14_0__snapshot_2_st_14_0
3: __snapshot_2_st_14_0__snapshot_2_st_14_0__snapshot_2_st_14_0__snapshot_2_st_14_0
4: __snapshot_2_st_14_0__snapshot_2_st_14_0__snapshot_2_st_14_0__snapshot_2_st_14_0

5.1.1.1.6 Fresh Variable Addition

The new variable __snapshot_2_rt_11_post is introduced. The transition formulas are extended as follows:

7: __snapshot_2_rt_11_postrt_11_postrt_11_post__snapshot_2_rt_11_post
9: __snapshot_2_rt_11_post__snapshot_2_rt_11_post__snapshot_2_rt_11_post__snapshot_2_rt_11_post
3: __snapshot_2_rt_11_post__snapshot_2_rt_11_post__snapshot_2_rt_11_post__snapshot_2_rt_11_post
4: __snapshot_2_rt_11_post__snapshot_2_rt_11_post__snapshot_2_rt_11_post__snapshot_2_rt_11_post

5.1.1.1.7 Fresh Variable Addition

The new variable __snapshot_2_rt_11_0 is introduced. The transition formulas are extended as follows:

7: __snapshot_2_rt_11_0rt_11_0rt_11_0__snapshot_2_rt_11_0
9: __snapshot_2_rt_11_0__snapshot_2_rt_11_0__snapshot_2_rt_11_0__snapshot_2_rt_11_0
3: __snapshot_2_rt_11_0__snapshot_2_rt_11_0__snapshot_2_rt_11_0__snapshot_2_rt_11_0
4: __snapshot_2_rt_11_0__snapshot_2_rt_11_0__snapshot_2_rt_11_0__snapshot_2_rt_11_0

5.1.1.1.8 Invariant Updates

The following invariants are asserted.

0: TRUE
1: TRUE
2: 1 − y_13_0 ≤ 0
3: 1 − y_13_0 ≤ 0
4: TRUE
2: 1 − y_13_0 ≤ 01 − __snapshot_2_x_15_0 + x_15_0 ≤ 01 − __snapshot_2_x_15_0 ≤ 01 − y_13_0 ≤ 0
3: 1 − __snapshot_2_x_15_0 + x_15_0 ≤ 01 − __snapshot_2_x_15_0 ≤ 01 − y_13_0 ≤ 0
2_var_snapshot: __snapshot_2_x_15_0 + x_15_0 ≤ 01 − __snapshot_2_x_15_0 + x_15_0y_13_0 ≤ 0y_13_0 ≤ 0
2*: 1 − __snapshot_2_x_15_0 + x_15_0 ≤ 01 − __snapshot_2_x_15_0 ≤ 01 − y_13_0 ≤ 0

The invariants are proved as follows.

IMPACT Invariant Proof

5.1.1.1.9 Transition Removal

We remove transition 9 using the following ranking functions, which are bounded by −1.

2: x_15_0
3: __snapshot_2_x_15_0
2_var_snapshot: __snapshot_2_x_15_0
2*: __snapshot_2_x_15_0
Hints:
7 distribute assertion
lexWeak[ [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0] ]
lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0] ]
9 lexStrict[ [1, 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, 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] ]
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, 1, 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, 1, 0, 0, 0, 0, 0, 0, 0] ]

5.1.1.1.10 Transition Removal

We remove transition 7 using the following ranking functions, which are bounded by −6.

2: −1
2_var_snapshot: −2
3: −3
2*: −4
Hints:
7 distribute assertion
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] ]
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] ]
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] ]
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] ]

5.1.1.1.11 Splitting Cut-Point Transitions

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

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