LTS Termination Proof

Input

Integer Transition System

Initial Location: l5, l4, l6, l3, l0, l2

Transitions: (pre-variables and post-variables)

l0	1	l1:	x1 = _oldX0HAT0 ∧ x2 = _oldX1HAT0 ∧ x3 = _oldX2HAT0 ∧ x4 = _oldX3HAT0 ∧ x5 = _x0HAT0 ∧ x6 = _x1HAT0 ∧ x1 = _oldX0HATpost ∧ x2 = _oldX1HATpost ∧ x3 = _oldX2HATpost ∧ x4 = _oldX3HATpost ∧ x5 = _x0HATpost ∧ x6 = _x1HATpost ∧ _x1HATpost = _oldX3HATpost ∧ _x0HATpost = _oldX2HATpost ∧ _oldX3HATpost = _oldX3HATpost ∧ _oldX2HATpost = _oldX2HATpost ∧ _oldX1HATpost = _x1HAT0 ∧ _oldX0HATpost = _x0HAT0
l0	2	l2:	x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x5 = _x4 ∧ x6 = _x5 ∧ x1 = _x6 ∧ x2 = _x7 ∧ x3 = _x8 ∧ x4 = _x9 ∧ x5 = _x10 ∧ x6 = _x11 ∧ _x3 = _x9 ∧ _x2 = _x8 ∧ _x11 = −1 + _x7 ∧ _x10 = −1 + _x6 ∧ _x7 = _x5 ∧ _x6 = _x4
l3	3	l1:	x1 = _x12 ∧ x2 = _x13 ∧ x3 = _x14 ∧ x4 = _x15 ∧ x5 = _x16 ∧ x6 = _x17 ∧ x1 = _x18 ∧ x2 = _x19 ∧ x3 = _x20 ∧ x4 = _x21 ∧ x5 = _x22 ∧ x6 = _x23 ∧ _x23 = _x21 ∧ _x22 = _x20 ∧ 0 ≤ _x18 ∧ _x18 ≤ 0 ∧ _x21 = _x21 ∧ _x20 = _x20 ∧ _x19 = _x17 ∧ _x18 = _x16
l3	4	l1:	x1 = _x24 ∧ x2 = _x25 ∧ x3 = _x26 ∧ x4 = _x27 ∧ x5 = _x28 ∧ x6 = _x29 ∧ x1 = _x30 ∧ x2 = _x31 ∧ x3 = _x32 ∧ x4 = _x33 ∧ x5 = _x34 ∧ x6 = _x35 ∧ _x35 = _x33 ∧ _x34 = _x32 ∧ 1 ≤ _x30 ∧ _x33 = _x33 ∧ _x32 = _x32 ∧ _x31 = _x29 ∧ _x30 = _x28
l3	5	l1:	x1 = _x36 ∧ x2 = _x37 ∧ x3 = _x38 ∧ x4 = _x39 ∧ x5 = _x40 ∧ x6 = _x41 ∧ x1 = _x42 ∧ x2 = _x43 ∧ x3 = _x44 ∧ x4 = _x45 ∧ x5 = _x46 ∧ x6 = _x47 ∧ _x47 = _x45 ∧ _x46 = _x44 ∧ 1 + _x42 ≤ 0 ∧ _x45 = _x45 ∧ _x44 = _x44 ∧ _x43 = _x41 ∧ _x42 = _x40
l4	6	l0:	x1 = _x48 ∧ x2 = _x49 ∧ x3 = _x50 ∧ x4 = _x51 ∧ x5 = _x52 ∧ x6 = _x53 ∧ x1 = _x54 ∧ x2 = _x55 ∧ x3 = _x56 ∧ x4 = _x57 ∧ x5 = _x58 ∧ x6 = _x59 ∧ _x51 = _x57 ∧ _x50 = _x56 ∧ _x59 = _x55 ∧ _x58 = _x54 ∧ 1 ≤ _x55 ∧ 1 ≤ _x54 ∧ _x55 = _x53 ∧ _x54 = _x52
l4	7	l3:	x1 = _x60 ∧ x2 = _x61 ∧ x3 = _x62 ∧ x4 = _x63 ∧ x5 = _x64 ∧ x6 = _x65 ∧ x1 = _x66 ∧ x2 = _x67 ∧ x3 = _x68 ∧ x4 = _x69 ∧ x5 = _x70 ∧ x6 = _x71 ∧ _x63 = _x69 ∧ _x62 = _x68 ∧ _x71 = _x67 ∧ _x70 = _x66 ∧ _x67 ≤ 0 ∧ _x67 = _x65 ∧ _x66 = _x64
l4	8	l3:	x1 = _x72 ∧ x2 = _x73 ∧ x3 = _x74 ∧ x4 = _x75 ∧ x5 = _x76 ∧ x6 = _x77 ∧ x1 = _x78 ∧ x2 = _x79 ∧ x3 = _x80 ∧ x4 = _x81 ∧ x5 = _x82 ∧ x6 = _x83 ∧ _x75 = _x81 ∧ _x74 = _x80 ∧ _x83 = _x79 ∧ _x82 = _x78 ∧ _x78 ≤ 0 ∧ _x79 = _x77 ∧ _x78 = _x76
l2	9	l4:	x1 = _x84 ∧ x2 = _x85 ∧ x3 = _x86 ∧ x4 = _x87 ∧ x5 = _x88 ∧ x6 = _x89 ∧ x1 = _x90 ∧ x2 = _x91 ∧ x3 = _x92 ∧ x4 = _x93 ∧ x5 = _x94 ∧ x6 = _x95 ∧ _x87 = _x93 ∧ _x86 = _x92 ∧ _x95 = _x91 ∧ _x94 = _x90 ∧ _x91 = _x89 ∧ _x90 = _x88
l5	10	l1:	x1 = _x96 ∧ x2 = _x97 ∧ x3 = _x98 ∧ x4 = _x99 ∧ x5 = _x100 ∧ x6 = _x101 ∧ x1 = _x102 ∧ x2 = _x103 ∧ x3 = _x104 ∧ x4 = _x105 ∧ x5 = _x106 ∧ x6 = _x107 ∧ _x101 = _x107 ∧ _x100 = _x106 ∧ _x99 = _x105 ∧ _x98 = _x104 ∧ _x97 = _x103 ∧ _x96 = _x102
l5	11	l0:	x1 = _x108 ∧ x2 = _x109 ∧ x3 = _x110 ∧ x4 = _x111 ∧ x5 = _x112 ∧ x6 = _x113 ∧ x1 = _x114 ∧ x2 = _x115 ∧ x3 = _x116 ∧ x4 = _x117 ∧ x5 = _x118 ∧ x6 = _x119 ∧ _x113 = _x119 ∧ _x112 = _x118 ∧ _x111 = _x117 ∧ _x110 = _x116 ∧ _x109 = _x115 ∧ _x108 = _x114
l5	12	l3:	x1 = _x120 ∧ x2 = _x121 ∧ x3 = _x122 ∧ x4 = _x123 ∧ x5 = _x124 ∧ x6 = _x125 ∧ x1 = _x126 ∧ x2 = _x127 ∧ x3 = _x128 ∧ x4 = _x129 ∧ x5 = _x130 ∧ x6 = _x131 ∧ _x125 = _x131 ∧ _x124 = _x130 ∧ _x123 = _x129 ∧ _x122 = _x128 ∧ _x121 = _x127 ∧ _x120 = _x126
l5	13	l4:	x1 = _x132 ∧ x2 = _x133 ∧ x3 = _x134 ∧ x4 = _x135 ∧ x5 = _x136 ∧ x6 = _x137 ∧ x1 = _x138 ∧ x2 = _x139 ∧ x3 = _x140 ∧ x4 = _x141 ∧ x5 = _x142 ∧ x6 = _x143 ∧ _x137 = _x143 ∧ _x136 = _x142 ∧ _x135 = _x141 ∧ _x134 = _x140 ∧ _x133 = _x139 ∧ _x132 = _x138
l5	14	l2:	x1 = _x144 ∧ x2 = _x145 ∧ x3 = _x146 ∧ x4 = _x147 ∧ x5 = _x148 ∧ x6 = _x149 ∧ x1 = _x150 ∧ x2 = _x151 ∧ x3 = _x152 ∧ x4 = _x153 ∧ x5 = _x154 ∧ x6 = _x155 ∧ _x149 = _x155 ∧ _x148 = _x154 ∧ _x147 = _x153 ∧ _x146 = _x152 ∧ _x145 = _x151 ∧ _x144 = _x150
l6	15	l5:	x1 = _x156 ∧ x2 = _x157 ∧ x3 = _x158 ∧ x4 = _x159 ∧ x5 = _x160 ∧ x6 = _x161 ∧ x1 = _x162 ∧ x2 = _x163 ∧ x3 = _x164 ∧ x4 = _x165 ∧ x5 = _x166 ∧ x6 = _x167 ∧ _x161 = _x167 ∧ _x160 = _x166 ∧ _x159 = _x165 ∧ _x158 = _x164 ∧ _x157 = _x163 ∧ _x156 = _x162

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

l5	l5	l5:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6
l4	l4	l4:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6
l6	l6	l6:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6
l3	l3	l3:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6
l0	l0	l0:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6
l2	l2	l2:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6

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 { l4, l0, l2 }.

2.1.1 Transition Removal

We remove transition 6 using the following ranking functions, which are bounded by 0.

l0:	−3 + x5 + x6
l2:	−1 + x5 + x6
l4:	−1 + x5 + x6

2.1.2 Transition Removal

We remove transitions 2, 9 using the following ranking functions, which are bounded by 0.

l0:	2
l2:	1
l4:	0

2.1.3 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 % (6 real / 0 unknown / 0 assumptions / 6 total proof steps)