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 = _oldX4HAT0 ∧ x6 = _oldX5HAT0 ∧ x7 = _x0HAT0 ∧ x8 = _x1HAT0 ∧ x9 = _x2HAT0 ∧ x1 = _oldX0HATpost ∧ x2 = _oldX1HATpost ∧ x3 = _oldX2HATpost ∧ x4 = _oldX3HATpost ∧ x5 = _oldX4HATpost ∧ x6 = _oldX5HATpost ∧ x7 = _x0HATpost ∧ x8 = _x1HATpost ∧ x9 = _x2HATpost ∧ _x2HATpost = _oldX5HATpost ∧ _x1HATpost = _oldX4HATpost ∧ _x0HATpost = _oldX3HATpost ∧ _oldX5HATpost = _oldX5HATpost ∧ _oldX4HATpost = _oldX4HATpost ∧ _oldX3HATpost = _oldX3HATpost ∧ _oldX2HATpost = _x2HAT0 ∧ _oldX1HATpost = _x1HAT0 ∧ _oldX0HATpost = _x0HAT0
l2	2	l3:	x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x5 = _x4 ∧ x6 = _x5 ∧ x7 = _x6 ∧ x8 = _x7 ∧ x9 = _x8 ∧ x1 = _x9 ∧ x2 = _x10 ∧ x3 = _x11 ∧ x4 = _x12 ∧ x5 = _x13 ∧ x6 = _x14 ∧ x7 = _x15 ∧ x8 = _x16 ∧ x9 = _x17 ∧ _x5 = _x14 ∧ _x4 = _x13 ∧ _x3 = _x12 ∧ _x17 = − _x10 + _x11 ∧ _x16 = _x10 ∧ _x15 = _x9 ∧ _x11 = _x8 ∧ _x10 = _x7 ∧ _x9 = _x6
l3	3	l0:	x1 = _x18 ∧ x2 = _x19 ∧ x3 = _x20 ∧ x4 = _x21 ∧ x5 = _x22 ∧ x6 = _x23 ∧ x7 = _x24 ∧ x8 = _x25 ∧ x9 = _x26 ∧ x1 = _x27 ∧ x2 = _x28 ∧ x3 = _x29 ∧ x4 = _x30 ∧ x5 = _x31 ∧ x6 = _x32 ∧ x7 = _x33 ∧ x8 = _x34 ∧ x9 = _x35 ∧ _x23 = _x32 ∧ _x22 = _x31 ∧ _x21 = _x30 ∧ _x35 = _x29 ∧ _x34 = _x28 ∧ _x33 = _x27 ∧ _x28 ≤ 0 ∧ _x29 = _x26 ∧ _x28 = _x25 ∧ _x27 = _x24
l3	4	l0:	x1 = _x36 ∧ x2 = _x37 ∧ x3 = _x38 ∧ x4 = _x39 ∧ x5 = _x40 ∧ x6 = _x41 ∧ x7 = _x42 ∧ x8 = _x43 ∧ x9 = _x44 ∧ x1 = _x45 ∧ x2 = _x46 ∧ x3 = _x47 ∧ x4 = _x48 ∧ x5 = _x49 ∧ x6 = _x50 ∧ x7 = _x51 ∧ x8 = _x52 ∧ x9 = _x53 ∧ _x41 = _x50 ∧ _x40 = _x49 ∧ _x39 = _x48 ∧ _x53 = _x47 ∧ _x52 = _x46 ∧ _x51 = _x45 ∧ 1 + _x47 ≤ _x46 ∧ _x47 = _x44 ∧ _x46 = _x43 ∧ _x45 = _x42
l3	5	l2:	x1 = _x54 ∧ x2 = _x55 ∧ x3 = _x56 ∧ x4 = _x57 ∧ x5 = _x58 ∧ x6 = _x59 ∧ x7 = _x60 ∧ x8 = _x61 ∧ x9 = _x62 ∧ x1 = _x63 ∧ x2 = _x64 ∧ x3 = _x65 ∧ x4 = _x66 ∧ x5 = _x67 ∧ x6 = _x68 ∧ x7 = _x69 ∧ x8 = _x70 ∧ x9 = _x71 ∧ _x59 = _x68 ∧ _x58 = _x67 ∧ _x57 = _x66 ∧ _x71 = _x65 ∧ _x70 = _x64 ∧ _x69 = _x63 ∧ 1 ≤ _x64 ∧ _x64 ≤ _x65 ∧ _x65 = _x62 ∧ _x64 = _x61 ∧ _x63 = _x60
l4	6	l3:	x1 = _x72 ∧ x2 = _x73 ∧ x3 = _x74 ∧ x4 = _x75 ∧ x5 = _x76 ∧ x6 = _x77 ∧ x7 = _x78 ∧ x8 = _x79 ∧ x9 = _x80 ∧ x1 = _x81 ∧ x2 = _x82 ∧ x3 = _x83 ∧ x4 = _x84 ∧ x5 = _x85 ∧ x6 = _x86 ∧ x7 = _x87 ∧ x8 = _x88 ∧ x9 = _x89 ∧ _x77 = _x86 ∧ _x76 = _x85 ∧ _x75 = _x84 ∧ _x89 = _x81 ∧ _x88 = _x82 ∧ _x87 = _x81 ∧ _x83 = _x80 ∧ _x82 = _x79 ∧ _x81 = _x78
l5	7	l4:	x1 = _x90 ∧ x2 = _x91 ∧ x3 = _x92 ∧ x4 = _x93 ∧ x5 = _x94 ∧ x6 = _x95 ∧ x7 = _x96 ∧ x8 = _x97 ∧ x9 = _x98 ∧ x1 = _x99 ∧ x2 = _x100 ∧ x3 = _x101 ∧ x4 = _x102 ∧ x5 = _x103 ∧ x6 = _x104 ∧ x7 = _x105 ∧ x8 = _x106 ∧ x9 = _x107 ∧ _x95 = _x104 ∧ _x94 = _x103 ∧ _x107 = _x102 ∧ _x106 = _x100 ∧ _x105 = _x99 ∧ _x102 = _x102 ∧ _x101 = _x98 ∧ _x100 = _x97 ∧ _x99 = _x96
l5	8	l1:	x1 = _x108 ∧ x2 = _x109 ∧ x3 = _x110 ∧ x4 = _x111 ∧ x5 = _x112 ∧ x6 = _x113 ∧ x7 = _x114 ∧ x8 = _x115 ∧ x9 = _x116 ∧ x1 = _x117 ∧ x2 = _x118 ∧ x3 = _x119 ∧ x4 = _x120 ∧ x5 = _x121 ∧ x6 = _x122 ∧ x7 = _x123 ∧ x8 = _x124 ∧ x9 = _x125 ∧ _x116 = _x125 ∧ _x115 = _x124 ∧ _x114 = _x123 ∧ _x113 = _x122 ∧ _x112 = _x121 ∧ _x111 = _x120 ∧ _x110 = _x119 ∧ _x109 = _x118 ∧ _x108 = _x117
l5	9	l0:	x1 = _x126 ∧ x2 = _x127 ∧ x3 = _x128 ∧ x4 = _x129 ∧ x5 = _x130 ∧ x6 = _x131 ∧ x7 = _x132 ∧ x8 = _x133 ∧ x9 = _x134 ∧ x1 = _x135 ∧ x2 = _x136 ∧ x3 = _x137 ∧ x4 = _x138 ∧ x5 = _x139 ∧ x6 = _x140 ∧ x7 = _x141 ∧ x8 = _x142 ∧ x9 = _x143 ∧ _x134 = _x143 ∧ _x133 = _x142 ∧ _x132 = _x141 ∧ _x131 = _x140 ∧ _x130 = _x139 ∧ _x129 = _x138 ∧ _x128 = _x137 ∧ _x127 = _x136 ∧ _x126 = _x135
l5	10	l2:	x1 = _x144 ∧ x2 = _x145 ∧ x3 = _x146 ∧ x4 = _x147 ∧ x5 = _x148 ∧ x6 = _x149 ∧ x7 = _x150 ∧ x8 = _x151 ∧ x9 = _x152 ∧ x1 = _x153 ∧ x2 = _x154 ∧ x3 = _x155 ∧ x4 = _x156 ∧ x5 = _x157 ∧ x6 = _x158 ∧ x7 = _x159 ∧ x8 = _x160 ∧ x9 = _x161 ∧ _x152 = _x161 ∧ _x151 = _x160 ∧ _x150 = _x159 ∧ _x149 = _x158 ∧ _x148 = _x157 ∧ _x147 = _x156 ∧ _x146 = _x155 ∧ _x145 = _x154 ∧ _x144 = _x153
l5	11	l3:	x1 = _x162 ∧ x2 = _x163 ∧ x3 = _x164 ∧ x4 = _x165 ∧ x5 = _x166 ∧ x6 = _x167 ∧ x7 = _x168 ∧ x8 = _x169 ∧ x9 = _x170 ∧ x1 = _x171 ∧ x2 = _x172 ∧ x3 = _x173 ∧ x4 = _x174 ∧ x5 = _x175 ∧ x6 = _x176 ∧ x7 = _x177 ∧ x8 = _x178 ∧ x9 = _x179 ∧ _x170 = _x179 ∧ _x169 = _x178 ∧ _x168 = _x177 ∧ _x167 = _x176 ∧ _x166 = _x175 ∧ _x165 = _x174 ∧ _x164 = _x173 ∧ _x163 = _x172 ∧ _x162 = _x171
l5	12	l4:	x1 = _x180 ∧ x2 = _x181 ∧ x3 = _x182 ∧ x4 = _x183 ∧ x5 = _x184 ∧ x6 = _x185 ∧ x7 = _x186 ∧ x8 = _x187 ∧ x9 = _x188 ∧ x1 = _x189 ∧ x2 = _x190 ∧ x3 = _x191 ∧ x4 = _x192 ∧ x5 = _x193 ∧ x6 = _x194 ∧ x7 = _x195 ∧ x8 = _x196 ∧ x9 = _x197 ∧ _x188 = _x197 ∧ _x187 = _x196 ∧ _x186 = _x195 ∧ _x185 = _x194 ∧ _x184 = _x193 ∧ _x183 = _x192 ∧ _x182 = _x191 ∧ _x181 = _x190 ∧ _x180 = _x189
l6	13	l5:	x1 = _x198 ∧ x2 = _x199 ∧ x3 = _x200 ∧ x4 = _x201 ∧ x5 = _x202 ∧ x6 = _x203 ∧ x7 = _x204 ∧ x8 = _x205 ∧ x9 = _x206 ∧ x1 = _x207 ∧ x2 = _x208 ∧ x3 = _x209 ∧ x4 = _x210 ∧ x5 = _x211 ∧ x6 = _x212 ∧ x7 = _x213 ∧ x8 = _x214 ∧ x9 = _x215 ∧ _x206 = _x215 ∧ _x205 = _x214 ∧ _x204 = _x213 ∧ _x203 = _x212 ∧ _x202 = _x211 ∧ _x201 = _x210 ∧ _x200 = _x209 ∧ _x199 = _x208 ∧ _x198 = _x207

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 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9
l4	l4	l4:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9
l6	l6	l6:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9
l3	l3	l3:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9
l0	l0	l0:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9
l2	l2	l2:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9

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 { l3, l2 }.

2.1.1 Transition Removal

We remove transition 5 using the following ranking functions, which are bounded by 0.

l2:	−1 + x8 + x9
l3:	−1 + 2⋅x8 + x9

2.1.2 Transition Removal

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

l2:	0
l3:	−1

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)