LTS Termination Proof

by T2Cert

Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

0: 1 − x_6_0 ≤ 0
1: Result_4_post ≤ 0____cil_tmp4_8_post ≤ 0____retres3_7_post ≤ 0____retres3_7_post ≤ 0Result_4_0 ≤ 0____cil_tmp4_8_0 ≤ 0____retres3_7_0 ≤ 0____retres3_7_0 ≤ 0
2: 1 − x_6_0 ≤ 0
3: TRUE
4: i_5_post ≤ 0i_5_post ≤ 0i_5_0 ≤ 0i_5_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:
0 7 0: x_6_0 + x_6_0 ≤ 0x_6_0x_6_0 ≤ 0i_5_post + i_5_post ≤ 0i_5_posti_5_post ≤ 0i_5_0 + i_5_0 ≤ 0i_5_0i_5_0 ≤ 0____retres3_7_post + ____retres3_7_post ≤ 0____retres3_7_post____retres3_7_post ≤ 0____retres3_7_0 + ____retres3_7_0 ≤ 0____retres3_7_0____retres3_7_0 ≤ 0____cil_tmp4_8_post + ____cil_tmp4_8_post ≤ 0____cil_tmp4_8_post____cil_tmp4_8_post ≤ 0____cil_tmp4_8_0 + ____cil_tmp4_8_0 ≤ 0____cil_tmp4_8_0____cil_tmp4_8_0 ≤ 0Result_4_post + Result_4_post ≤ 0Result_4_postResult_4_post ≤ 0Result_4_0 + Result_4_0 ≤ 0Result_4_0Result_4_0 ≤ 0
and for every transition t, a duplicate t is considered.

3 Transition Removal

We remove transitions 0, 3, 4, 5, 6 using the following ranking functions, which are bounded by −15.

5: 0
3: 0
4: 0
0: 0
2: 0
1: 0
5: −6
3: −7
4: −8
0: −9
2: −9
0_var_snapshot: −9
0*: −9
1: −13
Hints:
8 lexWeak[ [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] ]
2 lexWeak[ [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] ]
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] ]
4 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] ]
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] ]
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] ]

4 Location Addition

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

0* 10 0: x_6_0 + x_6_0 ≤ 0x_6_0x_6_0 ≤ 0i_5_post + i_5_post ≤ 0i_5_posti_5_post ≤ 0i_5_0 + i_5_0 ≤ 0i_5_0i_5_0 ≤ 0____retres3_7_post + ____retres3_7_post ≤ 0____retres3_7_post____retres3_7_post ≤ 0____retres3_7_0 + ____retres3_7_0 ≤ 0____retres3_7_0____retres3_7_0 ≤ 0____cil_tmp4_8_post + ____cil_tmp4_8_post ≤ 0____cil_tmp4_8_post____cil_tmp4_8_post ≤ 0____cil_tmp4_8_0 + ____cil_tmp4_8_0 ≤ 0____cil_tmp4_8_0____cil_tmp4_8_0 ≤ 0Result_4_post + Result_4_post ≤ 0Result_4_postResult_4_post ≤ 0Result_4_0 + Result_4_0 ≤ 0Result_4_0Result_4_0 ≤ 0

5 Location Addition

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

0 8 0_var_snapshot: x_6_0 + x_6_0 ≤ 0x_6_0x_6_0 ≤ 0i_5_post + i_5_post ≤ 0i_5_posti_5_post ≤ 0i_5_0 + i_5_0 ≤ 0i_5_0i_5_0 ≤ 0____retres3_7_post + ____retres3_7_post ≤ 0____retres3_7_post____retres3_7_post ≤ 0____retres3_7_0 + ____retres3_7_0 ≤ 0____retres3_7_0____retres3_7_0 ≤ 0____cil_tmp4_8_post + ____cil_tmp4_8_post ≤ 0____cil_tmp4_8_post____cil_tmp4_8_post ≤ 0____cil_tmp4_8_0 + ____cil_tmp4_8_0 ≤ 0____cil_tmp4_8_0____cil_tmp4_8_0 ≤ 0Result_4_post + Result_4_post ≤ 0Result_4_postResult_4_post ≤ 0Result_4_0 + Result_4_0 ≤ 0Result_4_0Result_4_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 { 0, 2, 0_var_snapshot, 0* }.

6.1.1 Transition Removal

We remove transition 1 using the following ranking functions, which are bounded by −404.

0: −2 − 4⋅i_5_0
2: −4⋅i_5_0
0_var_snapshot: −3 − 4⋅i_5_0
0*: −1 − 4⋅i_5_0
Hints:
8 lexWeak[ [0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
10 lexWeak[ [0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
1 lexStrict[ [0, 0, 0, 0, 0, 0, 4, 0, 4, 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] ]
2 lexWeak[ [0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

6.1.2 Transition Removal

We remove transitions 8, 2 using the following ranking functions, which are bounded by −1.

0: 0
2: 2
0_var_snapshot: −1
0*: 1
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] ]
10 lexWeak[ [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] ]

6.1.3 Transition Removal

We remove transition 10 using the following ranking functions, which are bounded by 0.

0: 0
2: 0
0_var_snapshot: 0
0*: x_6_0
Hints:
10 lexStrict[ [1, 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] ]

6.1.4 Splitting Cut-Point Transitions

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

6.1.4.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 7.

6.1.4.1.1 Splitting Cut-Point Transitions

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

Tool configuration

T2Cert