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 = _oldX6HAT0 ∧ x8 = _x0HAT0 ∧ x9 = _x1HAT0 ∧ x10 = _x2HAT0 ∧ x1 = _oldX0HATpost ∧ x2 = _oldX1HATpost ∧ x3 = _oldX2HATpost ∧ x4 = _oldX3HATpost ∧ x5 = _oldX4HATpost ∧ x6 = _oldX5HATpost ∧ x7 = _oldX6HATpost ∧ x8 = _x0HATpost ∧ x9 = _x1HATpost ∧ x10 = _x2HATpost ∧ _oldX6HAT0 = _oldX6HATpost ∧ _x2HATpost = _oldX5HATpost ∧ _x1HATpost = _oldX4HATpost ∧ _x0HATpost = _oldX3HATpost ∧ _oldX5HATpost = _oldX5HATpost ∧ _oldX4HATpost = _oldX4HATpost ∧ _oldX3HATpost = _oldX3HATpost ∧ _oldX2HATpost = _x2HAT0 ∧ _oldX1HATpost = _x1HAT0 ∧ _oldX0HATpost = _x0HAT0
l2	2	l1:	x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x5 = _x4 ∧ x6 = _x5 ∧ x7 = _x6 ∧ x8 = _x7 ∧ x9 = _x8 ∧ x10 = _x9 ∧ x1 = _x10 ∧ x2 = _x11 ∧ x3 = _x12 ∧ x4 = _x13 ∧ x5 = _x14 ∧ x6 = _x15 ∧ x7 = _x16 ∧ x8 = _x17 ∧ x9 = _x18 ∧ x10 = _x19 ∧ _x19 = _x15 ∧ _x18 = _x14 ∧ _x17 = _x13 ∧ 2 ≤ _x12 − 2⋅_x16 ∧ _x16 = _x16 ∧ _x15 = _x15 ∧ _x14 = _x14 ∧ _x13 = _x13 ∧ _x12 = _x9 ∧ _x11 = _x8 ∧ _x10 = _x7
l2	3	l1:	x1 = _x20 ∧ x2 = _x21 ∧ x3 = _x22 ∧ x4 = _x23 ∧ x5 = _x24 ∧ x6 = _x25 ∧ x7 = _x26 ∧ x8 = _x27 ∧ x9 = _x28 ∧ x10 = _x29 ∧ x1 = _x30 ∧ x2 = _x31 ∧ x3 = _x32 ∧ x4 = _x33 ∧ x5 = _x34 ∧ x6 = _x35 ∧ x7 = _x36 ∧ x8 = _x37 ∧ x9 = _x38 ∧ x10 = _x39 ∧ _x39 = _x35 ∧ _x38 = _x34 ∧ _x37 = _x33 ∧ 1 + _x32 − 2⋅_x36 ≤ 0 ∧ _x36 = _x36 ∧ _x35 = _x35 ∧ _x34 = _x34 ∧ _x33 = _x33 ∧ _x32 = _x29 ∧ _x31 = _x28 ∧ _x30 = _x27
l2	4	l3:	x1 = _x40 ∧ x2 = _x41 ∧ x3 = _x42 ∧ x4 = _x43 ∧ x5 = _x44 ∧ x6 = _x45 ∧ x7 = _x46 ∧ x8 = _x47 ∧ x9 = _x48 ∧ x10 = _x49 ∧ x1 = _x50 ∧ x2 = _x51 ∧ x3 = _x52 ∧ x4 = _x53 ∧ x5 = _x54 ∧ x6 = _x55 ∧ x7 = _x56 ∧ x8 = _x57 ∧ x9 = _x58 ∧ x10 = _x59 ∧ _x46 = _x56 ∧ _x45 = _x55 ∧ _x44 = _x54 ∧ _x59 = _x53 ∧ _x58 = 1 + _x51 ∧ _x57 = _x50 ∧ 1 + _x52 − 2⋅_x53 ≤ 2 ∧ 0 ≤ _x52 − 2⋅_x53 ∧ _x53 = _x53 ∧ _x52 = _x49 ∧ _x51 = _x48 ∧ _x50 = _x47
l3	5	l0:	x1 = _x60 ∧ x2 = _x61 ∧ x3 = _x62 ∧ x4 = _x63 ∧ x5 = _x64 ∧ x6 = _x65 ∧ x7 = _x66 ∧ x8 = _x67 ∧ x9 = _x68 ∧ x10 = _x69 ∧ x1 = _x70 ∧ x2 = _x71 ∧ x3 = _x72 ∧ x4 = _x73 ∧ x5 = _x74 ∧ x6 = _x75 ∧ x7 = _x76 ∧ x8 = _x77 ∧ x9 = _x78 ∧ x10 = _x79 ∧ _x66 = _x76 ∧ _x65 = _x75 ∧ _x64 = _x74 ∧ _x63 = _x73 ∧ _x79 = _x72 ∧ _x78 = _x71 ∧ _x77 = _x70 ∧ _x72 ≤ 1 ∧ _x72 = _x69 ∧ _x71 = _x68 ∧ _x70 = _x67
l3	6	l2:	x1 = _x80 ∧ x2 = _x81 ∧ x3 = _x82 ∧ x4 = _x83 ∧ x5 = _x84 ∧ x6 = _x85 ∧ x7 = _x86 ∧ x8 = _x87 ∧ x9 = _x88 ∧ x10 = _x89 ∧ x1 = _x90 ∧ x2 = _x91 ∧ x3 = _x92 ∧ x4 = _x93 ∧ x5 = _x94 ∧ x6 = _x95 ∧ x7 = _x96 ∧ x8 = _x97 ∧ x9 = _x98 ∧ x10 = _x99 ∧ _x86 = _x96 ∧ _x85 = _x95 ∧ _x84 = _x94 ∧ _x83 = _x93 ∧ _x99 = _x92 ∧ _x98 = _x91 ∧ _x97 = _x90 ∧ 2 ≤ _x92 ∧ _x92 = _x89 ∧ _x91 = _x88 ∧ _x90 = _x87
l4	7	l3:	x1 = _x100 ∧ x2 = _x101 ∧ x3 = _x102 ∧ x4 = _x103 ∧ x5 = _x104 ∧ x6 = _x105 ∧ x7 = _x106 ∧ x8 = _x107 ∧ x9 = _x108 ∧ x10 = _x109 ∧ x1 = _x110 ∧ x2 = _x111 ∧ x3 = _x112 ∧ x4 = _x113 ∧ x5 = _x114 ∧ x6 = _x115 ∧ x7 = _x116 ∧ x8 = _x117 ∧ x9 = _x118 ∧ x10 = _x119 ∧ _x106 = _x116 ∧ _x105 = _x115 ∧ _x104 = _x114 ∧ _x103 = _x113 ∧ _x119 = _x110 ∧ _x118 = 0 ∧ _x117 = _x110 ∧ _x112 = _x109 ∧ _x111 = _x108 ∧ _x110 = _x107
l5	8	l4:	x1 = _x120 ∧ x2 = _x121 ∧ x3 = _x122 ∧ x4 = _x123 ∧ x5 = _x124 ∧ x6 = _x125 ∧ x7 = _x126 ∧ x8 = _x127 ∧ x9 = _x128 ∧ x10 = _x129 ∧ x1 = _x130 ∧ x2 = _x131 ∧ x3 = _x132 ∧ x4 = _x133 ∧ x5 = _x134 ∧ x6 = _x135 ∧ x7 = _x136 ∧ x8 = _x137 ∧ x9 = _x138 ∧ x10 = _x139 ∧ _x126 = _x136 ∧ _x125 = _x135 ∧ _x139 = _x134 ∧ _x138 = _x133 ∧ _x137 = _x130 ∧ _x134 = _x134 ∧ _x133 = _x133 ∧ _x132 = _x129 ∧ _x131 = _x128 ∧ _x130 = _x127
l5	9	l1:	x1 = _x140 ∧ x2 = _x141 ∧ x3 = _x142 ∧ x4 = _x143 ∧ x5 = _x144 ∧ x6 = _x145 ∧ x7 = _x146 ∧ x8 = _x147 ∧ x9 = _x148 ∧ x10 = _x149 ∧ x1 = _x150 ∧ x2 = _x151 ∧ x3 = _x152 ∧ x4 = _x153 ∧ x5 = _x154 ∧ x6 = _x155 ∧ x7 = _x156 ∧ x8 = _x157 ∧ x9 = _x158 ∧ x10 = _x159 ∧ _x149 = _x159 ∧ _x148 = _x158 ∧ _x147 = _x157 ∧ _x146 = _x156 ∧ _x145 = _x155 ∧ _x144 = _x154 ∧ _x143 = _x153 ∧ _x142 = _x152 ∧ _x141 = _x151 ∧ _x140 = _x150
l5	10	l0:	x1 = _x160 ∧ x2 = _x161 ∧ x3 = _x162 ∧ x4 = _x163 ∧ x5 = _x164 ∧ x6 = _x165 ∧ x7 = _x166 ∧ x8 = _x167 ∧ x9 = _x168 ∧ x10 = _x169 ∧ x1 = _x170 ∧ x2 = _x171 ∧ x3 = _x172 ∧ x4 = _x173 ∧ x5 = _x174 ∧ x6 = _x175 ∧ x7 = _x176 ∧ x8 = _x177 ∧ x9 = _x178 ∧ x10 = _x179 ∧ _x169 = _x179 ∧ _x168 = _x178 ∧ _x167 = _x177 ∧ _x166 = _x176 ∧ _x165 = _x175 ∧ _x164 = _x174 ∧ _x163 = _x173 ∧ _x162 = _x172 ∧ _x161 = _x171 ∧ _x160 = _x170
l5	11	l2:	x1 = _x180 ∧ x2 = _x181 ∧ x3 = _x182 ∧ x4 = _x183 ∧ x5 = _x184 ∧ x6 = _x185 ∧ x7 = _x186 ∧ x8 = _x187 ∧ x9 = _x188 ∧ x10 = _x189 ∧ x1 = _x190 ∧ x2 = _x191 ∧ x3 = _x192 ∧ x4 = _x193 ∧ x5 = _x194 ∧ x6 = _x195 ∧ x7 = _x196 ∧ x8 = _x197 ∧ x9 = _x198 ∧ x10 = _x199 ∧ _x189 = _x199 ∧ _x188 = _x198 ∧ _x187 = _x197 ∧ _x186 = _x196 ∧ _x185 = _x195 ∧ _x184 = _x194 ∧ _x183 = _x193 ∧ _x182 = _x192 ∧ _x181 = _x191 ∧ _x180 = _x190
l5	12	l3:	x1 = _x200 ∧ x2 = _x201 ∧ x3 = _x202 ∧ x4 = _x203 ∧ x5 = _x204 ∧ x6 = _x205 ∧ x7 = _x206 ∧ x8 = _x207 ∧ x9 = _x208 ∧ x10 = _x209 ∧ x1 = _x210 ∧ x2 = _x211 ∧ x3 = _x212 ∧ x4 = _x213 ∧ x5 = _x214 ∧ x6 = _x215 ∧ x7 = _x216 ∧ x8 = _x217 ∧ x9 = _x218 ∧ x10 = _x219 ∧ _x209 = _x219 ∧ _x208 = _x218 ∧ _x207 = _x217 ∧ _x206 = _x216 ∧ _x205 = _x215 ∧ _x204 = _x214 ∧ _x203 = _x213 ∧ _x202 = _x212 ∧ _x201 = _x211 ∧ _x200 = _x210
l5	13	l4:	x1 = _x220 ∧ x2 = _x221 ∧ x3 = _x222 ∧ x4 = _x223 ∧ x5 = _x224 ∧ x6 = _x225 ∧ x7 = _x226 ∧ x8 = _x227 ∧ x9 = _x228 ∧ x10 = _x229 ∧ x1 = _x230 ∧ x2 = _x231 ∧ x3 = _x232 ∧ x4 = _x233 ∧ x5 = _x234 ∧ x6 = _x235 ∧ x7 = _x236 ∧ x8 = _x237 ∧ x9 = _x238 ∧ x10 = _x239 ∧ _x229 = _x239 ∧ _x228 = _x238 ∧ _x227 = _x237 ∧ _x226 = _x236 ∧ _x225 = _x235 ∧ _x224 = _x234 ∧ _x223 = _x233 ∧ _x222 = _x232 ∧ _x221 = _x231 ∧ _x220 = _x230
l6	14	l5:	x1 = _x240 ∧ x2 = _x241 ∧ x3 = _x242 ∧ x4 = _x243 ∧ x5 = _x244 ∧ x6 = _x245 ∧ x7 = _x246 ∧ x8 = _x247 ∧ x9 = _x248 ∧ x10 = _x249 ∧ x1 = _x250 ∧ x2 = _x251 ∧ x3 = _x252 ∧ x4 = _x253 ∧ x5 = _x254 ∧ x6 = _x255 ∧ x7 = _x256 ∧ x8 = _x257 ∧ x9 = _x258 ∧ x10 = _x259 ∧ _x249 = _x259 ∧ _x248 = _x258 ∧ _x247 = _x257 ∧ _x246 = _x256 ∧ _x245 = _x255 ∧ _x244 = _x254 ∧ _x243 = _x253 ∧ _x242 = _x252 ∧ _x241 = _x251 ∧ _x240 = _x250

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 ∧ x10 = x10
l4	l4	l4:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10
l6	l6	l6:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10
l3	l3	l3:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10
l0	l0	l0:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10
l2	l2	l2:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10

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

l2:	3⋅x10 + 1
l3:	6⋅x10

2.1.2 Transition Removal

We remove transition 4 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)