LTS Termination Proof

by T2Cert

Input

Integer Transition System

Initial Location: 4

Transitions: (pre-variables and post-variables)

0	0	1:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 10 − x_6_0 ≤ 0 ∧ ____retres3_7_post ≤ 0 ∧ − ____retres3_7_post ≤ 0 ∧ ____cil_tmp4_8_post − ____retres3_7_post ≤ 0 ∧ − ____cil_tmp4_8_post + ____retres3_7_post ≤ 0 ∧ Result_4_post − ____cil_tmp4_8_post ≤ 0 ∧ − Result_4_post + ____cil_tmp4_8_post ≤ 0 ∧ Result_4_0 − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_post ≤ 0 ∧ ____cil_tmp4_8_0 − ____cil_tmp4_8_post ≤ 0 ∧ − ____cil_tmp4_8_0 + ____cil_tmp4_8_post ≤ 0 ∧ ____retres3_7_0 − ____retres3_7_post ≤ 0 ∧ − ____retres3_7_0 + ____retres3_7_post ≤ 0 ∧ − x_6_0 + x_6_0 ≤ 0 ∧ x_6_0 − x_6_0 ≤ 0 ∧ − i_5_post + i_5_post ≤ 0 ∧ i_5_post − i_5_post ≤ 0 ∧ − i_5_0 + i_5_0 ≤ 0 ∧ i_5_0 − i_5_0 ≤ 0
0	1	1:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ −9 + x_6_0 ≤ 0 ∧ −100 + i_5_0 ≤ 0 ∧ ____retres3_7_post ≤ 0 ∧ − ____retres3_7_post ≤ 0 ∧ ____cil_tmp4_8_post − ____retres3_7_post ≤ 0 ∧ − ____cil_tmp4_8_post + ____retres3_7_post ≤ 0 ∧ Result_4_post − ____cil_tmp4_8_post ≤ 0 ∧ − Result_4_post + ____cil_tmp4_8_post ≤ 0 ∧ Result_4_0 − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_post ≤ 0 ∧ ____cil_tmp4_8_0 − ____cil_tmp4_8_post ≤ 0 ∧ − ____cil_tmp4_8_0 + ____cil_tmp4_8_post ≤ 0 ∧ ____retres3_7_0 − ____retres3_7_post ≤ 0 ∧ − ____retres3_7_0 + ____retres3_7_post ≤ 0 ∧ − x_6_0 + x_6_0 ≤ 0 ∧ x_6_0 − x_6_0 ≤ 0 ∧ − i_5_post + i_5_post ≤ 0 ∧ i_5_post − i_5_post ≤ 0 ∧ − i_5_0 + i_5_0 ≤ 0 ∧ i_5_0 − i_5_0 ≤ 0
0	2	2:	0 ≤ 0 ∧ 0 ≤ 0 ∧ −9 + x_6_0 ≤ 0 ∧ 101 − i_5_0 ≤ 0 ∧ 1 − i_5_0 + i_5_post ≤ 0 ∧ −1 + i_5_0 − i_5_post ≤ 0 ∧ i_5_0 − i_5_post ≤ 0 ∧ − i_5_0 + i_5_post ≤ 0 ∧ − x_6_0 + x_6_0 ≤ 0 ∧ x_6_0 − x_6_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 ≤ 0 ∧ − Result_4_post + Result_4_post ≤ 0 ∧ Result_4_post − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_0 ≤ 0 ∧ Result_4_0 − Result_4_0 ≤ 0
2	3	0:	− x_6_0 + x_6_0 ≤ 0 ∧ x_6_0 − x_6_0 ≤ 0 ∧ − i_5_post + i_5_post ≤ 0 ∧ i_5_post − i_5_post ≤ 0 ∧ − i_5_0 + i_5_0 ≤ 0 ∧ i_5_0 − i_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 ≤ 0 ∧ − Result_4_post + Result_4_post ≤ 0 ∧ Result_4_post − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_0 ≤ 0 ∧ Result_4_0 − Result_4_0 ≤ 0
3	4	0:	0 ≤ 0 ∧ 0 ≤ 0 ∧ −1000 + i_5_post ≤ 0 ∧ 1000 − i_5_post ≤ 0 ∧ i_5_0 − i_5_post ≤ 0 ∧ − i_5_0 + i_5_post ≤ 0 ∧ − x_6_0 + x_6_0 ≤ 0 ∧ x_6_0 − x_6_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 ≤ 0 ∧ − Result_4_post + Result_4_post ≤ 0 ∧ Result_4_post − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_0 ≤ 0 ∧ Result_4_0 − Result_4_0 ≤ 0
4	5	3:	− x_6_0 + x_6_0 ≤ 0 ∧ x_6_0 − x_6_0 ≤ 0 ∧ − i_5_post + i_5_post ≤ 0 ∧ i_5_post − i_5_post ≤ 0 ∧ − i_5_0 + i_5_0 ≤ 0 ∧ i_5_0 − i_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 ≤ 0 ∧ − Result_4_post + Result_4_post ≤ 0 ∧ Result_4_post − Result_4_post ≤ 0 ∧ − Result_4_0 + Result_4_0 ≤ 0 ∧ Result_4_0 − Result_4_0 ≤ 0

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 ≤ 0 ∧ Result_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

nodes (location) invariant:

0	(0)	TRUE
1	(1)	Result_4_post ≤ 0 ∧ ____cil_tmp4_8_post ≤ 0 ∧ ____retres3_7_post ≤ 0 ∧ − ____retres3_7_post ≤ 0 ∧ Result_4_0 ≤ 0 ∧ ____cil_tmp4_8_0 ≤ 0 ∧ ____retres3_7_0 ≤ 0 ∧ − ____retres3_7_0 ≤ 0
2	(2)	TRUE
3	(3)	TRUE
4	(4)	TRUE

initial node: 4
cover edges:
transition edges:

0 0 1

0 1 1

0 2 2

2 3 0

3 4 0

4 5 3

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

− x_6_0 + x_6_0 ≤ 0 ∧ x_6_0 − x_6_0 ≤ 0 ∧ − i_5_post + i_5_post ≤ 0 ∧ i_5_post − i_5_post ≤ 0 ∧ − i_5_0 + i_5_0 ≤ 0 ∧ i_5_0 − i_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 ≤ 0 ∧ − Result_4_post + Result_4_post ≤ 0 ∧ Result_4_post − Result_4_post ≤ 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 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

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 ≤ 0 ∧ x_6_0 − x_6_0 ≤ 0 ∧ − i_5_post + i_5_post ≤ 0 ∧ i_5_post − i_5_post ≤ 0 ∧ − i_5_0 + i_5_0 ≤ 0 ∧ i_5_0 − i_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 ≤ 0 ∧ − Result_4_post + Result_4_post ≤ 0 ∧ Result_4_post − Result_4_post ≤ 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.

0 7 0_var_snapshot: − x_6_0 + x_6_0 ≤ 0 ∧ x_6_0 − x_6_0 ≤ 0 ∧ − i_5_post + i_5_post ≤ 0 ∧ i_5_post − i_5_post ≤ 0 ∧ − i_5_0 + i_5_0 ≤ 0 ∧ i_5_0 − i_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 ≤ 0 ∧ − Result_4_post + Result_4_post ≤ 0 ∧ Result_4_post − Result_4_post ≤ 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 { 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

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

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

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.

Tool configuration

T2Cert

version: 1.0