LTS Termination Proof

Input

Integer Transition System

Initial Location: f97_0_random_GT, f162_0_power_GT', f155_0_main_InvokeMethod, f1_0_main_Load, f213_0_power_NE, f213_0_power_NE', f162_0_power_GT, __init

Transitions: (pre-variables and post-variables)

f1_0_main_Load	1	f162_0_power_GT:	x1 = _arg1 ∧ x2 = _arg2 ∧ x3 = _arg3 ∧ x4 = _arg4 ∧ x1 = _arg1P ∧ x2 = _arg2P ∧ x3 = _arg3P ∧ x4 = _arg4P ∧ 0 = _arg2P ∧ 0 = _arg1P ∧ 0 = _arg2 ∧ 0 ≤ _arg1 − 1
f1_0_main_Load	2	f97_0_random_GT:	x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x1 = _x4 ∧ x2 = _x5 ∧ x3 = _x6 ∧ x4 = _x7 ∧ 0 = _x5 ∧ 0 ≤ _x4 − 1 ∧ 0 ≤ _x − 1 ∧ 0 ≤ _x1 − 1 ∧ _x4 ≤ _x
f1_0_main_Load	3	f97_0_random_GT:	x1 = _x8 ∧ x2 = _x9 ∧ x3 = _x11 ∧ x4 = _x12 ∧ x1 = _x13 ∧ x2 = _x15 ∧ x3 = _x16 ∧ x4 = _x17 ∧ 0 ≤ _x13 − 1 ∧ 0 ≤ _x8 − 1 ∧ _x13 ≤ _x8 ∧ 0 ≤ _x9 − 1 ∧ −1 ≤ _x15 − 1
f97_0_random_GT	4	f162_0_power_GT:	x1 = _x18 ∧ x2 = _x20 ∧ x3 = _x21 ∧ x4 = _x22 ∧ x1 = _x23 ∧ x2 = _x24 ∧ x3 = _x25 ∧ x4 = _x26 ∧ 0 = _x24 ∧ _x20 = _x23 ∧ 0 ≤ _x18 − 1
f97_0_random_GT	5	f155_0_main_InvokeMethod:	x1 = _x27 ∧ x2 = _x28 ∧ x3 = _x29 ∧ x4 = _x30 ∧ x1 = _x31 ∧ x2 = _x32 ∧ x3 = _x33 ∧ x4 = _x34 ∧ _x31 ≤ _x27 ∧ 1 ≤ _x35 − 1 ∧ 0 ≤ _x27 − 1 ∧ 0 ≤ _x31 − 1 ∧ _x28 = _x32 ∧ 0 = _x33
f97_0_random_GT	6	f155_0_main_InvokeMethod:	x1 = _x36 ∧ x2 = _x38 ∧ x3 = _x39 ∧ x4 = _x40 ∧ x1 = _x42 ∧ x2 = _x43 ∧ x3 = _x44 ∧ x4 = _x45 ∧ 1 ≤ _x46 − 1 ∧ −1 ≤ _x44 − 1 ∧ _x42 ≤ _x36 ∧ 0 ≤ _x36 − 1 ∧ 0 ≤ _x42 − 1 ∧ _x38 = _x43
f155_0_main_InvokeMethod	7	f162_0_power_GT:	x1 = _x48 ∧ x2 = _x49 ∧ x3 = _x50 ∧ x4 = _x51 ∧ x1 = _x52 ∧ x2 = _x53 ∧ x3 = _x54 ∧ x4 = _x55 ∧ 0 ≤ _x48 − 1 ∧ 1 ≤ _x56 − 1 ∧ _x49 = _x52 ∧ _x50 = _x53
f162_0_power_GT	8	f162_0_power_GT':	x1 = _x57 ∧ x2 = _x58 ∧ x3 = _x59 ∧ x4 = _x60 ∧ x1 = _x61 ∧ x2 = _x62 ∧ x3 = _x63 ∧ x4 = _x64 ∧ _x58 = _x62 ∧ _x57 = _x61 ∧ 1 ≤ _x58 − 1 ∧ _x57 ≤ 1
f162_0_power_GT'	9	f213_0_power_NE:	x1 = _x65 ∧ x2 = _x66 ∧ x3 = _x67 ∧ x4 = _x68 ∧ x1 = _x69 ∧ x2 = _x70 ∧ x3 = _x71 ∧ x4 = _x72 ∧ 1 ≤ _x66 − 1 ∧ _x65 ≤ 1 ∧ _x66 − 2⋅_x73 ≤ 1 ∧ 0 ≤ _x66 − 2⋅_x73 ∧ _x65 = _x69 ∧ _x66 = _x70 ∧ _x66 − 2⋅_x73 = _x71
f162_0_power_GT	10	f162_0_power_GT':	x1 = _x74 ∧ x2 = _x75 ∧ x3 = _x76 ∧ x4 = _x77 ∧ x1 = _x78 ∧ x2 = _x79 ∧ x3 = _x80 ∧ x4 = _x81 ∧ _x75 = _x79 ∧ _x74 = _x78 ∧ 1 ≤ _x75 − 1 ∧ 2 ≤ _x74 − 1
f162_0_power_GT'	11	f213_0_power_NE:	x1 = _x82 ∧ x2 = _x83 ∧ x3 = _x84 ∧ x4 = _x85 ∧ x1 = _x86 ∧ x2 = _x87 ∧ x3 = _x88 ∧ x4 = _x89 ∧ 1 ≤ _x83 − 1 ∧ 2 ≤ _x82 − 1 ∧ _x83 − 2⋅_x90 ≤ 1 ∧ 0 ≤ _x83 − 2⋅_x90 ∧ _x82 = _x86 ∧ _x83 = _x87 ∧ _x83 − 2⋅_x90 = _x88
f213_0_power_NE	12	f213_0_power_NE':	x1 = _x91 ∧ x2 = _x92 ∧ x3 = _x93 ∧ x4 = _x94 ∧ x1 = _x95 ∧ x2 = _x96 ∧ x3 = _x97 ∧ x4 = _x98 ∧ 1 ≤ _x92 − 1 ∧ 0 ≤ _x99 − 1 ∧ _x99 ≤ _x92 − 1 ∧ 0 = _x93 ∧ _x91 = _x95 ∧ _x92 = _x96 ∧ 0 = _x97 ∧ _x94 = _x98
f213_0_power_NE'	13	f162_0_power_GT:	x1 = _x100 ∧ x2 = _x101 ∧ x3 = _x102 ∧ x4 = _x103 ∧ x1 = _x104 ∧ x2 = _x105 ∧ x3 = _x106 ∧ x4 = _x107 ∧ _x100 = _x104 ∧ 0 = _x102 ∧ 0 ≤ _x101 − 2⋅_x105 ∧ _x101 − 2⋅_x105 ≤ 1 ∧ 0 ≤ _x105 − 1 ∧ _x105 ≤ _x101 − 1 ∧ 1 ≤ _x101 − 1
f213_0_power_NE	14	f162_0_power_GT:	x1 = _x108 ∧ x2 = _x109 ∧ x3 = _x110 ∧ x4 = _x111 ∧ x1 = _x112 ∧ x2 = _x113 ∧ x3 = _x114 ∧ x4 = _x115 ∧ _x109 − 1 = _x113 ∧ _x108 = _x112 ∧ 1 = _x110 ∧ _x109 − 1 ≤ _x109 − 1 ∧ 1 ≤ _x109 − 1
__init	15	f1_0_main_Load:	x1 = _x116 ∧ x2 = _x117 ∧ x3 = _x118 ∧ x4 = _x119 ∧ x1 = _x120 ∧ x2 = _x121 ∧ x3 = _x122 ∧ x4 = _x123 ∧ 0 ≤ 0

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

f97_0_random_GT	f97_0_random_GT	f97_0_random_GT:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4
f162_0_power_GT'	f162_0_power_GT'	f162_0_power_GT':	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4
f155_0_main_InvokeMethod	f155_0_main_InvokeMethod	f155_0_main_InvokeMethod:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4
f1_0_main_Load	f1_0_main_Load	f1_0_main_Load:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4
f213_0_power_NE	f213_0_power_NE	f213_0_power_NE:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4
f213_0_power_NE'	f213_0_power_NE'	f213_0_power_NE':	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4
f162_0_power_GT	f162_0_power_GT	f162_0_power_GT:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4
__init	__init	__init:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4

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

2 SCC Decomposition

We consider subproblems for each of the 1 SCC(s) of the program graph.

2.1 SCC Subproblem 1/1

Here we consider the SCC { f162_0_power_GT', f213_0_power_NE, f213_0_power_NE', f162_0_power_GT }.

2.1.1 Transition Removal

We remove transitions 8, 13, 12, 9, 10, 14, 11 using the following ranking functions, which are bounded by 0.

f162_0_power_GT:	4⋅x2 + 2
f162_0_power_GT':	4⋅x2 + 1
f213_0_power_NE':	2⋅x2 + 3
f213_0_power_NE:	4⋅x2

2.1.2 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 % (5 real / 0 unknown / 0 assumptions / 5 total proof steps)