LTS Termination Proof

Input

Integer Transition System

Initial Location: f218_0_times_Return, f469_0_times_InvokeMethod, f484_0_plus_LE, f127_0_times_NE, f1_0_main_Load, __init

Transitions: (pre-variables and post-variables)

f1_0_main_Load	1	f127_0_times_NE:	x1 = _arg1 ∧ x2 = _arg2 ∧ x3 = _arg3 ∧ x1 = _arg1P ∧ x2 = _arg2P ∧ x3 = _arg3P ∧ 0 ≤ _arg1 − 1 ∧ −1 ≤ _arg1P − 1 ∧ 1 ≤ _arg2 − 1 ∧ −1 ≤ _arg2P − 1
f127_0_times_NE	2	f127_0_times_NE:	x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x1 = _x3 ∧ x2 = _x4 ∧ x3 = _x5 ∧ _x1 − 1 = _x4 ∧ _x = _x3 ∧ _x1 − 1 ≤ _x1 − 1 ∧ 0 ≤ _x1 − 1
f127_0_times_NE	3	f469_0_times_InvokeMethod:	x1 = _x6 ∧ x2 = _x7 ∧ x3 = _x8 ∧ x1 = _x9 ∧ x2 = _x10 ∧ x3 = _x11 ∧ _x6 = _x11 ∧ 0 = _x10 ∧ 1 = _x9 ∧ 1 = _x7
f127_0_times_NE	4	f469_0_times_InvokeMethod:	x1 = _x12 ∧ x2 = _x13 ∧ x3 = _x14 ∧ x1 = _x15 ∧ x2 = _x16 ∧ x3 = _x17 ∧ 0 = _x17 ∧ 0 = _x16 ∧ _x13 = _x15 ∧ 0 = _x12 ∧ _x13 − 1 ≤ _x13 − 1 ∧ 0 ≤ _x13 − 1
f127_0_times_NE	5	f469_0_times_InvokeMethod:	x1 = _x18 ∧ x2 = _x19 ∧ x3 = _x20 ∧ x1 = _x21 ∧ x2 = _x22 ∧ x3 = _x23 ∧ _x18 = _x23 ∧ _x19 = _x21 ∧ _x19 − 1 ≤ _x19 − 1 ∧ 0 ≤ _x19 − 1
f127_0_times_NE	6	f469_0_times_InvokeMethod:	x1 = _x24 ∧ x2 = _x25 ∧ x3 = _x26 ∧ x1 = _x27 ∧ x2 = _x28 ∧ x3 = _x29 ∧ 0 = _x29 ∧ _x25 = _x27 ∧ 0 = _x24 ∧ _x25 − 1 ≤ _x25 − 1 ∧ 0 ≤ _x25 − 1
f218_0_times_Return	7	f469_0_times_InvokeMethod:	x1 = _x30 ∧ x2 = _x31 ∧ x3 = _x32 ∧ x1 = _x33 ∧ x2 = _x34 ∧ x3 = _x35 ∧ _x31 = _x35 ∧ _x32 = _x34 ∧ _x30 = _x33
f469_0_times_InvokeMethod	8	f484_0_plus_LE:	x1 = _x36 ∧ x2 = _x37 ∧ x3 = _x38 ∧ x1 = _x39 ∧ x2 = _x40 ∧ x3 = _x41 ∧ _x38 = _x40 ∧ _x37 = _x39 ∧ 0 ≤ _x36 − 1
f484_0_plus_LE	9	f484_0_plus_LE:	x1 = _x42 ∧ x2 = _x43 ∧ x3 = _x44 ∧ x1 = _x45 ∧ x2 = _x46 ∧ x3 = _x47 ∧ _x43 − 1 = _x46 ∧ _x42 = _x45 ∧ _x43 − 1 ≤ _x43 − 1 ∧ 0 ≤ _x43 − 1
f484_0_plus_LE	10	f484_0_plus_LE:	x1 = _x48 ∧ x2 = _x49 ∧ x3 = _x50 ∧ x1 = _x51 ∧ x2 = _x52 ∧ x3 = _x53 ∧ _x49 = _x52 ∧ _x48 − 1 = _x51 ∧ 0 ≤ _x48 − 1 ∧ _x48 − 1 ≤ _x48 − 1 ∧ _x49 ≤ 0
__init	11	f1_0_main_Load:	x1 = _x54 ∧ x2 = _x55 ∧ x3 = _x56 ∧ x1 = _x57 ∧ x2 = _x58 ∧ x3 = _x59 ∧ 0 ≤ 0

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

f218_0_times_Return	f218_0_times_Return	f218_0_times_Return:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3
f469_0_times_InvokeMethod	f469_0_times_InvokeMethod	f469_0_times_InvokeMethod:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3
f484_0_plus_LE	f484_0_plus_LE	f484_0_plus_LE:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3
f127_0_times_NE	f127_0_times_NE	f127_0_times_NE:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3
f1_0_main_Load	f1_0_main_Load	f1_0_main_Load:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3
__init	__init	__init:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3

and for every transition t, a duplicate t is considered.

2 SCC Decomposition

We consider subproblems for each of the 2 SCC(s) of the program graph.

2.1 SCC Subproblem 1/2

Here we consider the SCC { f127_0_times_NE }.

2.1.1 Transition Removal

We remove transition 2 using the following ranking functions, which are bounded by 0.

f127_0_times_NE:

2.1.2 Trivial Cooperation Program

There are no more "sharp" transitions in the cooperation program. Hence the cooperation termination is proved.

2.2 SCC Subproblem 2/2

Here we consider the SCC { f484_0_plus_LE }.

2.2.1 Transition Removal

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

f484_0_plus_LE:

−1 + x1

2.2.2 Transition Removal

We remove transition 9 using the following ranking functions, which are bounded by 0.

f484_0_plus_LE:

2.2.3 Trivial Cooperation Program

There are no more "sharp" transitions in the cooperation program. Hence the cooperation termination is proved.

Tool configuration

AProVE

version: AProVE Commit ID: unknown
strategy: Statistics for single proof: 100.00 % (9 real / 0 unknown / 0 assumptions / 9 total proof steps)