# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: l5, l4, l7, l6, l10, l1, l3, l0, l2, l9
• Transitions: (pre-variables and post-variables)  l0 1 l1: x1 = ___const_10HAT0 ∧ x2 = ___const_285HAT0 ∧ x3 = ___const_35HAT0 ∧ x4 = _acc12HAT0 ∧ x5 = _acc_length11HAT0 ∧ x6 = _coef_len210HAT0 ∧ x7 = _coef_len6HAT0 ∧ x8 = _i8HAT0 ∧ x9 = _in_len4HAT0 ∧ x10 = _j9HAT0 ∧ x11 = _scale7HAT0 ∧ x1 = ___const_10HATpost ∧ x2 = ___const_285HATpost ∧ x3 = ___const_35HATpost ∧ x4 = _acc12HATpost ∧ x5 = _acc_length11HATpost ∧ x6 = _coef_len210HATpost ∧ x7 = _coef_len6HATpost ∧ x8 = _i8HATpost ∧ x9 = _in_len4HATpost ∧ x10 = _j9HATpost ∧ x11 = _scale7HATpost ∧ _scale7HAT0 = _scale7HATpost ∧ _j9HAT0 = _j9HATpost ∧ _in_len4HAT0 = _in_len4HATpost ∧ _i8HAT0 = _i8HATpost ∧ _coef_len6HAT0 = _coef_len6HATpost ∧ _coef_len210HAT0 = _coef_len210HATpost ∧ _acc_length11HAT0 = _acc_length11HATpost ∧ _acc12HAT0 = _acc12HATpost ∧ ___const_35HAT0 = ___const_35HATpost ∧ ___const_285HAT0 = ___const_285HATpost ∧ ___const_10HAT0 = ___const_10HATpost l2 2 l0: x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x5 = _x4 ∧ x6 = _x5 ∧ x7 = _x6 ∧ x8 = _x7 ∧ x9 = _x8 ∧ x10 = _x9 ∧ x11 = _x10 ∧ x1 = _x11 ∧ x2 = _x12 ∧ x3 = _x13 ∧ x4 = _x14 ∧ x5 = _x15 ∧ x6 = _x16 ∧ x7 = _x17 ∧ x8 = _x18 ∧ x9 = _x19 ∧ x10 = _x20 ∧ x11 = _x21 ∧ _x10 = _x21 ∧ _x9 = _x20 ∧ _x8 = _x19 ∧ _x7 = _x18 ∧ _x6 = _x17 ∧ _x5 = _x16 ∧ _x4 = _x15 ∧ _x3 = _x14 ∧ _x2 = _x13 ∧ _x1 = _x12 ∧ _x = _x11 ∧ _x6 ≤ _x4 l2 3 l0: x1 = _x22 ∧ x2 = _x23 ∧ x3 = _x24 ∧ x4 = _x25 ∧ x5 = _x26 ∧ x6 = _x27 ∧ x7 = _x28 ∧ x8 = _x29 ∧ x9 = _x30 ∧ x10 = _x31 ∧ x11 = _x32 ∧ x1 = _x33 ∧ x2 = _x34 ∧ x3 = _x35 ∧ x4 = _x36 ∧ x5 = _x37 ∧ x6 = _x38 ∧ x7 = _x39 ∧ x8 = _x40 ∧ x9 = _x41 ∧ x10 = _x42 ∧ x11 = _x43 ∧ _x32 = _x43 ∧ _x31 = _x42 ∧ _x30 = _x41 ∧ _x29 = _x40 ∧ _x28 = _x39 ∧ _x27 = _x38 ∧ _x25 = _x36 ∧ _x24 = _x35 ∧ _x23 = _x34 ∧ _x22 = _x33 ∧ _x37 = 1 + _x26 ∧ 1 + _x26 ≤ _x28 l1 4 l3: x1 = _x44 ∧ x2 = _x45 ∧ x3 = _x46 ∧ x4 = _x47 ∧ x5 = _x48 ∧ x6 = _x49 ∧ x7 = _x50 ∧ x8 = _x51 ∧ x9 = _x52 ∧ x10 = _x53 ∧ x11 = _x54 ∧ x1 = _x55 ∧ x2 = _x56 ∧ x3 = _x57 ∧ x4 = _x58 ∧ x5 = _x59 ∧ x6 = _x60 ∧ x7 = _x61 ∧ x8 = _x62 ∧ x9 = _x63 ∧ x10 = _x64 ∧ x11 = _x65 ∧ _x54 = _x65 ∧ _x53 = _x64 ∧ _x52 = _x63 ∧ _x50 = _x61 ∧ _x49 = _x60 ∧ _x48 = _x59 ∧ _x47 = _x58 ∧ _x46 = _x57 ∧ _x45 = _x56 ∧ _x44 = _x55 ∧ _x62 = 1 + _x51 l4 5 l2: x1 = _x66 ∧ x2 = _x67 ∧ x3 = _x68 ∧ x4 = _x69 ∧ x5 = _x70 ∧ x6 = _x71 ∧ x7 = _x72 ∧ x8 = _x73 ∧ x9 = _x74 ∧ x10 = _x75 ∧ x11 = _x76 ∧ x1 = _x77 ∧ x2 = _x78 ∧ x3 = _x79 ∧ x4 = _x80 ∧ x5 = _x81 ∧ x6 = _x82 ∧ x7 = _x83 ∧ x8 = _x84 ∧ x9 = _x85 ∧ x10 = _x86 ∧ x11 = _x87 ∧ _x76 = _x87 ∧ _x75 = _x86 ∧ _x74 = _x85 ∧ _x73 = _x84 ∧ _x72 = _x83 ∧ _x71 = _x82 ∧ _x70 = _x81 ∧ _x69 = _x80 ∧ _x68 = _x79 ∧ _x67 = _x78 ∧ _x66 = _x77 l4 6 l1: x1 = _x88 ∧ x2 = _x89 ∧ x3 = _x90 ∧ x4 = _x91 ∧ x5 = _x92 ∧ x6 = _x93 ∧ x7 = _x94 ∧ x8 = _x95 ∧ x9 = _x96 ∧ x10 = _x97 ∧ x11 = _x98 ∧ x1 = _x99 ∧ x2 = _x100 ∧ x3 = _x101 ∧ x4 = _x102 ∧ x5 = _x103 ∧ x6 = _x104 ∧ x7 = _x105 ∧ x8 = _x106 ∧ x9 = _x107 ∧ x10 = _x108 ∧ x11 = _x109 ∧ _x98 = _x109 ∧ _x97 = _x108 ∧ _x96 = _x107 ∧ _x95 = _x106 ∧ _x94 = _x105 ∧ _x93 = _x104 ∧ _x91 = _x102 ∧ _x90 = _x101 ∧ _x89 = _x100 ∧ _x88 = _x99 ∧ _x103 = −1 + _x92 l3 7 l5: x1 = _x110 ∧ x2 = _x111 ∧ x3 = _x112 ∧ x4 = _x113 ∧ x5 = _x114 ∧ x6 = _x115 ∧ x7 = _x116 ∧ x8 = _x117 ∧ x9 = _x118 ∧ x10 = _x119 ∧ x11 = _x120 ∧ x1 = _x121 ∧ x2 = _x122 ∧ x3 = _x123 ∧ x4 = _x124 ∧ x5 = _x125 ∧ x6 = _x126 ∧ x7 = _x127 ∧ x8 = _x128 ∧ x9 = _x129 ∧ x10 = _x130 ∧ x11 = _x131 ∧ _x120 = _x131 ∧ _x119 = _x130 ∧ _x118 = _x129 ∧ _x117 = _x128 ∧ _x116 = _x127 ∧ _x115 = _x126 ∧ _x114 = _x125 ∧ _x113 = _x124 ∧ _x112 = _x123 ∧ _x111 = _x122 ∧ _x110 = _x121 l6 8 l4: x1 = _x132 ∧ x2 = _x133 ∧ x3 = _x134 ∧ x4 = _x135 ∧ x5 = _x136 ∧ x6 = _x137 ∧ x7 = _x138 ∧ x8 = _x139 ∧ x9 = _x140 ∧ x10 = _x141 ∧ x11 = _x142 ∧ x1 = _x143 ∧ x2 = _x144 ∧ x3 = _x145 ∧ x4 = _x146 ∧ x5 = _x147 ∧ x6 = _x148 ∧ x7 = _x149 ∧ x8 = _x150 ∧ x9 = _x151 ∧ x10 = _x152 ∧ x11 = _x153 ∧ _x142 = _x153 ∧ _x141 = _x152 ∧ _x140 = _x151 ∧ _x139 = _x150 ∧ _x138 = _x149 ∧ _x137 = _x148 ∧ _x136 = _x147 ∧ _x135 = _x146 ∧ _x134 = _x145 ∧ _x133 = _x144 ∧ _x132 = _x143 ∧ _x136 ≤ _x141 l6 9 l7: x1 = _x154 ∧ x2 = _x155 ∧ x3 = _x156 ∧ x4 = _x157 ∧ x5 = _x158 ∧ x6 = _x159 ∧ x7 = _x160 ∧ x8 = _x161 ∧ x9 = _x162 ∧ x10 = _x163 ∧ x11 = _x164 ∧ x1 = _x165 ∧ x2 = _x166 ∧ x3 = _x167 ∧ x4 = _x168 ∧ x5 = _x169 ∧ x6 = _x170 ∧ x7 = _x171 ∧ x8 = _x172 ∧ x9 = _x173 ∧ x10 = _x174 ∧ x11 = _x175 ∧ _x164 = _x175 ∧ _x162 = _x173 ∧ _x161 = _x172 ∧ _x160 = _x171 ∧ _x159 = _x170 ∧ _x158 = _x169 ∧ _x156 = _x167 ∧ _x155 = _x166 ∧ _x154 = _x165 ∧ _x174 = 1 + _x163 ∧ _x168 = _x168 ∧ 1 + _x163 ≤ _x158 l7 10 l6: x1 = _x176 ∧ x2 = _x177 ∧ x3 = _x178 ∧ x4 = _x179 ∧ x5 = _x180 ∧ x6 = _x181 ∧ x7 = _x182 ∧ x8 = _x183 ∧ x9 = _x184 ∧ x10 = _x185 ∧ x11 = _x186 ∧ x1 = _x187 ∧ x2 = _x188 ∧ x3 = _x189 ∧ x4 = _x190 ∧ x5 = _x191 ∧ x6 = _x192 ∧ x7 = _x193 ∧ x8 = _x194 ∧ x9 = _x195 ∧ x10 = _x196 ∧ x11 = _x197 ∧ _x186 = _x197 ∧ _x185 = _x196 ∧ _x184 = _x195 ∧ _x183 = _x194 ∧ _x182 = _x193 ∧ _x181 = _x192 ∧ _x180 = _x191 ∧ _x179 = _x190 ∧ _x178 = _x189 ∧ _x177 = _x188 ∧ _x176 = _x187 l5 11 l8: x1 = _x198 ∧ x2 = _x199 ∧ x3 = _x200 ∧ x4 = _x201 ∧ x5 = _x202 ∧ x6 = _x203 ∧ x7 = _x204 ∧ x8 = _x205 ∧ x9 = _x206 ∧ x10 = _x207 ∧ x11 = _x208 ∧ x1 = _x209 ∧ x2 = _x210 ∧ x3 = _x211 ∧ x4 = _x212 ∧ x5 = _x213 ∧ x6 = _x214 ∧ x7 = _x215 ∧ x8 = _x216 ∧ x9 = _x217 ∧ x10 = _x218 ∧ x11 = _x219 ∧ _x208 = _x219 ∧ _x207 = _x218 ∧ _x206 = _x217 ∧ _x205 = _x216 ∧ _x204 = _x215 ∧ _x203 = _x214 ∧ _x202 = _x213 ∧ _x201 = _x212 ∧ _x200 = _x211 ∧ _x199 = _x210 ∧ _x198 = _x209 ∧ _x206 ≤ _x205 l5 12 l7: x1 = _x220 ∧ x2 = _x221 ∧ x3 = _x222 ∧ x4 = _x223 ∧ x5 = _x224 ∧ x6 = _x225 ∧ x7 = _x226 ∧ x8 = _x227 ∧ x9 = _x228 ∧ x10 = _x229 ∧ x11 = _x230 ∧ x1 = _x231 ∧ x2 = _x232 ∧ x3 = _x233 ∧ x4 = _x234 ∧ x5 = _x235 ∧ x6 = _x236 ∧ x7 = _x237 ∧ x8 = _x238 ∧ x9 = _x239 ∧ x10 = _x240 ∧ x11 = _x241 ∧ _x230 = _x241 ∧ _x228 = _x239 ∧ _x227 = _x238 ∧ _x226 = _x237 ∧ _x225 = _x236 ∧ _x224 = _x235 ∧ _x222 = _x233 ∧ _x221 = _x232 ∧ _x220 = _x231 ∧ _x240 = 1 ∧ _x234 = _x234 ∧ 1 + _x227 ≤ _x228 l9 13 l3: x1 = _x242 ∧ x2 = _x243 ∧ x3 = _x244 ∧ x4 = _x245 ∧ x5 = _x246 ∧ x6 = _x247 ∧ x7 = _x248 ∧ x8 = _x249 ∧ x9 = _x250 ∧ x10 = _x251 ∧ x11 = _x252 ∧ x1 = _x253 ∧ x2 = _x254 ∧ x3 = _x255 ∧ x4 = _x256 ∧ x5 = _x257 ∧ x6 = _x258 ∧ x7 = _x259 ∧ x8 = _x260 ∧ x9 = _x261 ∧ x10 = _x262 ∧ x11 = _x263 ∧ _x251 = _x262 ∧ _x245 = _x256 ∧ _x244 = _x255 ∧ _x243 = _x254 ∧ _x242 = _x253 ∧ _x260 = 0 ∧ _x257 = _x258 ∧ _x258 = _x258 ∧ _x263 = _x243 ∧ _x259 = _x244 ∧ _x261 = _x242 l10 14 l9: x1 = _x264 ∧ x2 = _x265 ∧ x3 = _x266 ∧ x4 = _x267 ∧ x5 = _x268 ∧ x6 = _x269 ∧ x7 = _x270 ∧ x8 = _x271 ∧ x9 = _x272 ∧ x10 = _x273 ∧ x11 = _x274 ∧ x1 = _x275 ∧ x2 = _x276 ∧ x3 = _x277 ∧ x4 = _x278 ∧ x5 = _x279 ∧ x6 = _x280 ∧ x7 = _x281 ∧ x8 = _x282 ∧ x9 = _x283 ∧ x10 = _x284 ∧ x11 = _x285 ∧ _x274 = _x285 ∧ _x273 = _x284 ∧ _x272 = _x283 ∧ _x271 = _x282 ∧ _x270 = _x281 ∧ _x269 = _x280 ∧ _x268 = _x279 ∧ _x267 = _x278 ∧ _x266 = _x277 ∧ _x265 = _x276 ∧ _x264 = _x275

## 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 ∧ x11 = x11 l4 l4 l4: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 ∧ x11 = x11 l7 l7 l7: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 ∧ x11 = x11 l6 l6 l6: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 ∧ x11 = x11 l10 l10 l10: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 ∧ x11 = x11 l1 l1 l1: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 ∧ x11 = x11 l3 l3 l3: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 ∧ x11 = x11 l0 l0 l0: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 ∧ x11 = x11 l2 l2 l2: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 ∧ x11 = x11 l9 l9 l9: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 ∧ x11 = x11
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 { l5, l4, l7, l6, l1, l3, l0, l2 }.

### 2.1.1 Transition Removal

We remove transition 12 using the following ranking functions, which are bounded by 0.

 l0: −2 − x8 + x9 l1: −2 − x8 + x9 l2: −2 − x8 + x9 l4: −2 − x8 + x9 l6: −2 − x8 + x9 l7: −2 − x8 + x9 l5: −1 − x8 + x9 l3: −1 − x8 + x9

### 2.1.2 Transition Removal

We remove transitions 1, 3, 2, 5, 8, 7, 4, 6 using the following ranking functions, which are bounded by 0.

 l0: 2 l1: 1 l2: 3 l4: 4 l6: 5 l7: 5 l3: 0 l5: −1

### 2.1.3 Transition Removal

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

 l7: 2⋅x5 − 2⋅x10 + 1 l6: 2⋅x5 − 2⋅x10

### 2.1.4 Transition Removal

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

 l7: 0 l6: −1

### 2.1.5 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 % (8 real / 0 unknown / 0 assumptions / 8 total proof steps)