LTS Termination Proof

by T2Cert

Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

0: TRUE
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: TRUE
3: TRUE
4: TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
0 6 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, 1, 4, 5 using the following ranking functions, which are bounded by −13.

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

4 Location Addition

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

0* 9 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 7 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 2 using the following ranking functions, which are bounded by 400.

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:
7 lexWeak[ [0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
9 lexWeak[ [0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
2 lexStrict[ [0, 0, 0, 0, 4, 0, 4, 0, 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] ]
3 lexWeak[ [0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

6.1.2 Transition Removal

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

0: −1
2: 1
0_var_snapshot: −2
0*: 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] ]
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] ]
3 lexWeak[ [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 3 using the following ranking functions, which are bounded by −1.

0: 0
2: 0
0_var_snapshot: 0
0*: −1
Hints:
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] ]

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 6.

6.1.4.1.1 Splitting Cut-Point Transitions

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

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