LTS Termination Proof

by T2Cert

Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

1: −400 + maxRetries_9_post ≤ 0 ∧ 400 − maxRetries_9_post ≤ 0 ∧ −400 + maxRetries_9_0 ≤ 0 ∧ 400 − maxRetries_9_0 ≤ 0
3: −400 + maxRetries_9_post ≤ 0 ∧ 400 − maxRetries_9_post ≤ 0 ∧ −400 + maxRetries_9_0 ≤ 0 ∧ 400 − maxRetries_9_0 ≤ 0
4: −400 + maxRetries_9_post ≤ 0 ∧ 400 − maxRetries_9_post ≤ 0 ∧ −400 + maxRetries_9_0 ≤ 0 ∧ 400 − maxRetries_9_0 ≤ 0
5: −400 + maxRetries_9_post ≤ 0 ∧ 400 − maxRetries_9_post ≤ 0 ∧ −400 + maxRetries_9_0 ≤ 0 ∧ 400 − maxRetries_9_0 ≤ 0
6: −400 + maxRetries_9_post ≤ 0 ∧ 400 − maxRetries_9_post ≤ 0 ∧ 400 − maxRetries_9_0 ≤ 0 ∧ selected_11_0 ≤ 0 ∧ − selected_11_0 ≤ 0 ∧ Result_4_post ≤ 0 ∧ ____cil_tmp6_12_post ≤ 0 ∧ Result_4_0 ≤ 0 ∧ ____cil_tmp6_12_0 ≤ 0
7: TRUE
8: TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
3 10 3: − x_5_0 + x_5_0 ≤ 0 ∧ x_5_0 − x_5_0 ≤ 0 ∧ − selected_11_post + selected_11_post ≤ 0 ∧ selected_11_post − selected_11_post ≤ 0 ∧ − selected_11_0 + selected_11_0 ≤ 0 ∧ selected_11_0 − selected_11_0 ≤ 0 ∧ − retryCount_10_post + retryCount_10_post ≤ 0 ∧ retryCount_10_post − retryCount_10_post ≤ 0 ∧ − retryCount_10_0 + retryCount_10_0 ≤ 0 ∧ retryCount_10_0 − retryCount_10_0 ≤ 0 ∧ − maxRetries_9_post + maxRetries_9_post ≤ 0 ∧ maxRetries_9_post − maxRetries_9_post ≤ 0 ∧ − maxRetries_9_0 + maxRetries_9_0 ≤ 0 ∧ maxRetries_9_0 − maxRetries_9_0 ≤ 0 ∧ − ____cil_tmp6_12_post + ____cil_tmp6_12_post ≤ 0 ∧ ____cil_tmp6_12_post − ____cil_tmp6_12_post ≤ 0 ∧ − ____cil_tmp6_12_0 + ____cil_tmp6_12_0 ≤ 0 ∧ ____cil_tmp6_12_0 − ____cil_tmp6_12_0 ≤ 0 ∧ − ____cil_tmp2_6_post + ____cil_tmp2_6_post ≤ 0 ∧ ____cil_tmp2_6_post − ____cil_tmp2_6_post ≤ 0 ∧ − ____cil_tmp2_6_0 + ____cil_tmp2_6_0 ≤ 0 ∧ ____cil_tmp2_6_0 − ____cil_tmp2_6_0 ≤ 0 ∧ − Result_4_post + Result_4_post ≤ 0 ∧ Result_4_post − Result_4_post ≤ 0 ∧ − Result_4_1 + Result_4_1 ≤ 0 ∧ Result_4_1 − Result_4_1 ≤ 0 ∧ − Result_4_0 + Result_4_0 ≤ 0 ∧ Result_4_0 − Result_4_0 ≤ 0
and for every transition t, a duplicate t is considered.

3 Transition Removal

We remove transitions 2, 3, 4, 5, 8, 9 using the following ranking functions, which are bounded by −17.

8: 0
7: 0
1: 0
3: 0
4: 0
5: 0
6: 0
8: −7
7: −8
1: −9
3: −9
3_var_snapshot: −9
3*: −9
4: −13
5: −14
6: −15

4 Location Addition

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

3* 13 3: − x_5_0 + x_5_0 ≤ 0 ∧ x_5_0 − x_5_0 ≤ 0 ∧ − selected_11_post + selected_11_post ≤ 0 ∧ selected_11_post − selected_11_post ≤ 0 ∧ − selected_11_0 + selected_11_0 ≤ 0 ∧ selected_11_0 − selected_11_0 ≤ 0 ∧ − retryCount_10_post + retryCount_10_post ≤ 0 ∧ retryCount_10_post − retryCount_10_post ≤ 0 ∧ − retryCount_10_0 + retryCount_10_0 ≤ 0 ∧ retryCount_10_0 − retryCount_10_0 ≤ 0 ∧ − maxRetries_9_post + maxRetries_9_post ≤ 0 ∧ maxRetries_9_post − maxRetries_9_post ≤ 0 ∧ − maxRetries_9_0 + maxRetries_9_0 ≤ 0 ∧ maxRetries_9_0 − maxRetries_9_0 ≤ 0 ∧ − ____cil_tmp6_12_post + ____cil_tmp6_12_post ≤ 0 ∧ ____cil_tmp6_12_post − ____cil_tmp6_12_post ≤ 0 ∧ − ____cil_tmp6_12_0 + ____cil_tmp6_12_0 ≤ 0 ∧ ____cil_tmp6_12_0 − ____cil_tmp6_12_0 ≤ 0 ∧ − ____cil_tmp2_6_post + ____cil_tmp2_6_post ≤ 0 ∧ ____cil_tmp2_6_post − ____cil_tmp2_6_post ≤ 0 ∧ − ____cil_tmp2_6_0 + ____cil_tmp2_6_0 ≤ 0 ∧ ____cil_tmp2_6_0 − ____cil_tmp2_6_0 ≤ 0 ∧ − Result_4_post + Result_4_post ≤ 0 ∧ Result_4_post − Result_4_post ≤ 0 ∧ − Result_4_1 + Result_4_1 ≤ 0 ∧ Result_4_1 − Result_4_1 ≤ 0 ∧ − Result_4_0 + Result_4_0 ≤ 0 ∧ Result_4_0 − Result_4_0 ≤ 0

5 Location Addition

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

3 11 3_var_snapshot: − x_5_0 + x_5_0 ≤ 0 ∧ x_5_0 − x_5_0 ≤ 0 ∧ − selected_11_post + selected_11_post ≤ 0 ∧ selected_11_post − selected_11_post ≤ 0 ∧ − selected_11_0 + selected_11_0 ≤ 0 ∧ selected_11_0 − selected_11_0 ≤ 0 ∧ − retryCount_10_post + retryCount_10_post ≤ 0 ∧ retryCount_10_post − retryCount_10_post ≤ 0 ∧ − retryCount_10_0 + retryCount_10_0 ≤ 0 ∧ retryCount_10_0 − retryCount_10_0 ≤ 0 ∧ − maxRetries_9_post + maxRetries_9_post ≤ 0 ∧ maxRetries_9_post − maxRetries_9_post ≤ 0 ∧ − maxRetries_9_0 + maxRetries_9_0 ≤ 0 ∧ maxRetries_9_0 − maxRetries_9_0 ≤ 0 ∧ − ____cil_tmp6_12_post + ____cil_tmp6_12_post ≤ 0 ∧ ____cil_tmp6_12_post − ____cil_tmp6_12_post ≤ 0 ∧ − ____cil_tmp6_12_0 + ____cil_tmp6_12_0 ≤ 0 ∧ ____cil_tmp6_12_0 − ____cil_tmp6_12_0 ≤ 0 ∧ − ____cil_tmp2_6_post + ____cil_tmp2_6_post ≤ 0 ∧ ____cil_tmp2_6_post − ____cil_tmp2_6_post ≤ 0 ∧ − ____cil_tmp2_6_0 + ____cil_tmp2_6_0 ≤ 0 ∧ ____cil_tmp2_6_0 − ____cil_tmp2_6_0 ≤ 0 ∧ − Result_4_post + Result_4_post ≤ 0 ∧ Result_4_post − Result_4_post ≤ 0 ∧ − Result_4_1 + Result_4_1 ≤ 0 ∧ Result_4_1 − Result_4_1 ≤ 0 ∧ − Result_4_0 + Result_4_0 ≤ 0 ∧ Result_4_0 − Result_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 { 1, 3, 3_var_snapshot, 3* }.

6.1.1 Transition Removal

We remove transition 6 using the following ranking functions, which are bounded by −160798.

1: − maxRetries_9_0 − 403⋅retryCount_10_0
3: 1 − 403⋅retryCount_10_0
3_var_snapshot: −403⋅retryCount_10_0
3*: 2 − 403⋅retryCount_10_0

6.1.2 Transition Removal

We remove transitions 11, 13, 7 using the following ranking functions, which are bounded by −1.

1: 2
3: 0
3_var_snapshot: −1
3*: 1

6.1.3 Splitting Cut-Point Transitions

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

6.1.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 10.

6.1.3.1.1 Splitting Cut-Point Transitions

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

Tool configuration

T2Cert