LTS Termination Proof

by T2Cert

Input

Integer Transition System

Initial Location: 5

Transitions: (pre-variables and post-variables)

0	0	1:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 101 − 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	1	2:	0 ≤ 0 ∧ 0 ≤ 0 ∧ −100 + i_5_0 ≤ 0 ∧ 1 − x_6_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	2	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	3	4:	0 ≤ 0 ∧ 0 ≤ 0 ∧ i_5_post ≤ 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
4	4	1:	0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ −100 + i_5_0 ≤ 0 ∧ 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
4	5	0:	0 ≤ 0 ∧ 0 ≤ 0 ∧ −100 + i_5_0 ≤ 0 ∧ 1 − x_6_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
5	6	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:	1 − x_6_0 ≤ 0
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:	1 − x_6_0 ≤ 0
3:	TRUE
4:	i_5_post ≤ 0 ∧ − i_5_post ≤ 0 ∧ i_5_0 ≤ 0 ∧ − i_5_0 ≤ 0
5:	TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

nodes (location) invariant:

0	(0)	1 − x_6_0 ≤ 0
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)	1 − x_6_0 ≤ 0
3	(3)	TRUE
4	(4)	i_5_post ≤ 0 ∧ − i_5_post ≤ 0 ∧ i_5_0 ≤ 0 ∧ − i_5_0 ≤ 0
5	(5)	TRUE

initial node: 5
cover edges:
transition edges:

0 0 1

0 1 2

2 2 0

3 3 4

4 4 1

4 5 0

5 6 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, 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

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 ≤ 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 8 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 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

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

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

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

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