# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: l4, l7, l10, l6, l11, l8, l1, l3, l0, l12, l2, l9
• Transitions: (pre-variables and post-variables)  l0 1 l1: x1 = _N6HAT0 ∧ x2 = _NHAT0 ∧ x3 = _i8HAT0 ∧ x4 = _iHAT0 ∧ x5 = _j7HAT0 ∧ x6 = _min9HAT0 ∧ x7 = _t10HAT0 ∧ x8 = _tmpHAT0 ∧ x1 = _N6HATpost ∧ x2 = _NHATpost ∧ x3 = _i8HATpost ∧ x4 = _iHATpost ∧ x5 = _j7HATpost ∧ x6 = _min9HATpost ∧ x7 = _t10HATpost ∧ x8 = _tmpHATpost ∧ _tmpHAT0 = _tmpHATpost ∧ _t10HAT0 = _t10HATpost ∧ _min9HAT0 = _min9HATpost ∧ _j7HAT0 = _j7HATpost ∧ _i8HAT0 = _i8HATpost ∧ _iHAT0 = _iHATpost ∧ _N6HAT0 = _N6HATpost ∧ _NHAT0 = _NHATpost l2 2 l3: x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x5 = _x4 ∧ x6 = _x5 ∧ x7 = _x6 ∧ x8 = _x7 ∧ x1 = _x8 ∧ x2 = _x9 ∧ x3 = _x10 ∧ x4 = _x11 ∧ x5 = _x12 ∧ x6 = _x13 ∧ x7 = _x14 ∧ x8 = _x15 ∧ _x7 = _x15 ∧ _x6 = _x14 ∧ _x5 = _x13 ∧ _x4 = _x12 ∧ _x2 = _x10 ∧ _x3 = _x11 ∧ _x = _x8 ∧ _x1 = _x9 l4 3 l5: x1 = _x16 ∧ x2 = _x17 ∧ x3 = _x18 ∧ x4 = _x19 ∧ x5 = _x20 ∧ x6 = _x21 ∧ x7 = _x22 ∧ x8 = _x23 ∧ x1 = _x24 ∧ x2 = _x25 ∧ x3 = _x26 ∧ x4 = _x27 ∧ x5 = _x28 ∧ x6 = _x29 ∧ x7 = _x30 ∧ x8 = _x31 ∧ _x23 = _x31 ∧ _x22 = _x30 ∧ _x21 = _x29 ∧ _x20 = _x28 ∧ _x18 = _x26 ∧ _x19 = _x27 ∧ _x16 = _x24 ∧ _x17 = _x25 ∧ −1 + _x16 ≤ _x20 l4 4 l6: x1 = _x32 ∧ x2 = _x33 ∧ x3 = _x34 ∧ x4 = _x35 ∧ x5 = _x36 ∧ x6 = _x37 ∧ x7 = _x38 ∧ x8 = _x39 ∧ x1 = _x40 ∧ x2 = _x41 ∧ x3 = _x42 ∧ x4 = _x43 ∧ x5 = _x44 ∧ x6 = _x45 ∧ x7 = _x46 ∧ x8 = _x47 ∧ _x39 = _x47 ∧ _x38 = _x46 ∧ _x37 = _x45 ∧ _x34 = _x42 ∧ _x35 = _x43 ∧ _x32 = _x40 ∧ _x33 = _x41 ∧ _x44 = 1 + _x36 ∧ 1 + _x36 ≤ −1 + _x32 l7 5 l8: x1 = _x48 ∧ x2 = _x49 ∧ x3 = _x50 ∧ x4 = _x51 ∧ x5 = _x52 ∧ x6 = _x53 ∧ x7 = _x54 ∧ x8 = _x55 ∧ x1 = _x56 ∧ x2 = _x57 ∧ x3 = _x58 ∧ x4 = _x59 ∧ x5 = _x60 ∧ x6 = _x61 ∧ x7 = _x62 ∧ x8 = _x63 ∧ _x55 = _x63 ∧ _x54 = _x62 ∧ _x53 = _x61 ∧ _x52 = _x60 ∧ _x50 = _x58 ∧ _x51 = _x59 ∧ _x48 = _x56 ∧ _x49 = _x57 l9 6 l7: x1 = _x64 ∧ x2 = _x65 ∧ x3 = _x66 ∧ x4 = _x67 ∧ x5 = _x68 ∧ x6 = _x69 ∧ x7 = _x70 ∧ x8 = _x71 ∧ x1 = _x72 ∧ x2 = _x73 ∧ x3 = _x74 ∧ x4 = _x75 ∧ x5 = _x76 ∧ x6 = _x77 ∧ x7 = _x78 ∧ x8 = _x79 ∧ _x71 = _x79 ∧ _x70 = _x78 ∧ _x69 = _x77 ∧ _x68 = _x76 ∧ _x67 = _x75 ∧ _x64 = _x72 ∧ _x65 = _x73 ∧ _x74 = 1 + _x66 l10 7 l9: x1 = _x80 ∧ x2 = _x81 ∧ x3 = _x82 ∧ x4 = _x83 ∧ x5 = _x84 ∧ x6 = _x85 ∧ x7 = _x86 ∧ x8 = _x87 ∧ x1 = _x88 ∧ x2 = _x89 ∧ x3 = _x90 ∧ x4 = _x91 ∧ x5 = _x92 ∧ x6 = _x93 ∧ x7 = _x94 ∧ x8 = _x95 ∧ _x87 = _x95 ∧ _x86 = _x94 ∧ _x84 = _x92 ∧ _x82 = _x90 ∧ _x83 = _x91 ∧ _x80 = _x88 ∧ _x81 = _x89 ∧ _x93 = _x82 l10 8 l9: x1 = _x96 ∧ x2 = _x97 ∧ x3 = _x98 ∧ x4 = _x99 ∧ x5 = _x100 ∧ x6 = _x101 ∧ x7 = _x102 ∧ x8 = _x103 ∧ x1 = _x104 ∧ x2 = _x105 ∧ x3 = _x106 ∧ x4 = _x107 ∧ x5 = _x108 ∧ x6 = _x109 ∧ x7 = _x110 ∧ x8 = _x111 ∧ _x103 = _x111 ∧ _x102 = _x110 ∧ _x101 = _x109 ∧ _x100 = _x108 ∧ _x98 = _x106 ∧ _x99 = _x107 ∧ _x96 = _x104 ∧ _x97 = _x105 l8 9 l2: x1 = _x112 ∧ x2 = _x113 ∧ x3 = _x114 ∧ x4 = _x115 ∧ x5 = _x116 ∧ x6 = _x117 ∧ x7 = _x118 ∧ x8 = _x119 ∧ x1 = _x120 ∧ x2 = _x121 ∧ x3 = _x122 ∧ x4 = _x123 ∧ x5 = _x124 ∧ x6 = _x125 ∧ x7 = _x126 ∧ x8 = _x127 ∧ _x119 = _x127 ∧ _x117 = _x125 ∧ _x114 = _x122 ∧ _x115 = _x123 ∧ _x112 = _x120 ∧ _x113 = _x121 ∧ _x124 = 1 + _x116 ∧ _x126 = _x126 ∧ _x112 ≤ _x114 l8 10 l10: x1 = _x128 ∧ x2 = _x129 ∧ x3 = _x130 ∧ x4 = _x131 ∧ x5 = _x132 ∧ x6 = _x133 ∧ x7 = _x134 ∧ x8 = _x135 ∧ x1 = _x136 ∧ x2 = _x137 ∧ x3 = _x138 ∧ x4 = _x139 ∧ x5 = _x140 ∧ x6 = _x141 ∧ x7 = _x142 ∧ x8 = _x143 ∧ _x135 = _x143 ∧ _x134 = _x142 ∧ _x133 = _x141 ∧ _x132 = _x140 ∧ _x130 = _x138 ∧ _x131 = _x139 ∧ _x128 = _x136 ∧ _x129 = _x137 ∧ 1 + _x130 ≤ _x128 l6 11 l4: x1 = _x144 ∧ x2 = _x145 ∧ x3 = _x146 ∧ x4 = _x147 ∧ x5 = _x148 ∧ x6 = _x149 ∧ x7 = _x150 ∧ x8 = _x151 ∧ x1 = _x152 ∧ x2 = _x153 ∧ x3 = _x154 ∧ x4 = _x155 ∧ x5 = _x156 ∧ x6 = _x157 ∧ x7 = _x158 ∧ x8 = _x159 ∧ _x151 = _x159 ∧ _x150 = _x158 ∧ _x149 = _x157 ∧ _x148 = _x156 ∧ _x146 = _x154 ∧ _x147 = _x155 ∧ _x144 = _x152 ∧ _x145 = _x153 l3 12 l6: x1 = _x160 ∧ x2 = _x161 ∧ x3 = _x162 ∧ x4 = _x163 ∧ x5 = _x164 ∧ x6 = _x165 ∧ x7 = _x166 ∧ x8 = _x167 ∧ x1 = _x168 ∧ x2 = _x169 ∧ x3 = _x170 ∧ x4 = _x171 ∧ x5 = _x172 ∧ x6 = _x173 ∧ x7 = _x174 ∧ x8 = _x175 ∧ _x167 = _x175 ∧ _x166 = _x174 ∧ _x165 = _x173 ∧ _x162 = _x170 ∧ _x163 = _x171 ∧ _x160 = _x168 ∧ _x161 = _x169 ∧ _x172 = 0 ∧ −1 + _x160 ≤ _x164 l3 13 l7: x1 = _x176 ∧ x2 = _x177 ∧ x3 = _x178 ∧ x4 = _x179 ∧ x5 = _x180 ∧ x6 = _x181 ∧ x7 = _x182 ∧ x8 = _x183 ∧ x1 = _x184 ∧ x2 = _x185 ∧ x3 = _x186 ∧ x4 = _x187 ∧ x5 = _x188 ∧ x6 = _x189 ∧ x7 = _x190 ∧ x8 = _x191 ∧ _x183 = _x191 ∧ _x182 = _x190 ∧ _x180 = _x188 ∧ _x179 = _x187 ∧ _x176 = _x184 ∧ _x177 = _x185 ∧ _x186 = 1 + _x180 ∧ _x189 = _x180 ∧ 1 + _x180 ≤ −1 + _x176 l1 14 l2: x1 = _x192 ∧ x2 = _x193 ∧ x3 = _x194 ∧ x4 = _x195 ∧ x5 = _x196 ∧ x6 = _x197 ∧ x7 = _x198 ∧ x8 = _x199 ∧ x1 = _x200 ∧ x2 = _x201 ∧ x3 = _x202 ∧ x4 = _x203 ∧ x5 = _x204 ∧ x6 = _x205 ∧ x7 = _x206 ∧ x8 = _x207 ∧ _x199 = _x207 ∧ _x198 = _x206 ∧ _x197 = _x205 ∧ _x194 = _x202 ∧ _x195 = _x203 ∧ _x193 = _x201 ∧ _x204 = 0 ∧ _x200 = _x193 ∧ _x193 ≤ _x195 l1 15 l0: x1 = _x208 ∧ x2 = _x209 ∧ x3 = _x210 ∧ x4 = _x211 ∧ x5 = _x212 ∧ x6 = _x213 ∧ x7 = _x214 ∧ x8 = _x215 ∧ x1 = _x216 ∧ x2 = _x217 ∧ x3 = _x218 ∧ x4 = _x219 ∧ x5 = _x220 ∧ x6 = _x221 ∧ x7 = _x222 ∧ x8 = _x223 ∧ _x215 = _x223 ∧ _x214 = _x222 ∧ _x213 = _x221 ∧ _x212 = _x220 ∧ _x210 = _x218 ∧ _x208 = _x216 ∧ _x209 = _x217 ∧ _x219 = 1 + _x211 ∧ 1 + _x211 ≤ _x209 l11 16 l0: x1 = _x224 ∧ x2 = _x225 ∧ x3 = _x226 ∧ x4 = _x227 ∧ x5 = _x228 ∧ x6 = _x229 ∧ x7 = _x230 ∧ x8 = _x231 ∧ x1 = _x232 ∧ x2 = _x233 ∧ x3 = _x234 ∧ x4 = _x235 ∧ x5 = _x236 ∧ x6 = _x237 ∧ x7 = _x238 ∧ x8 = _x239 ∧ _x230 = _x238 ∧ _x229 = _x237 ∧ _x228 = _x236 ∧ _x226 = _x234 ∧ _x224 = _x232 ∧ _x225 = _x233 ∧ _x235 = 0 ∧ _x239 = _x239 l12 17 l11: x1 = _x240 ∧ x2 = _x241 ∧ x3 = _x242 ∧ x4 = _x243 ∧ x5 = _x244 ∧ x6 = _x245 ∧ x7 = _x246 ∧ x8 = _x247 ∧ x1 = _x248 ∧ x2 = _x249 ∧ x3 = _x250 ∧ x4 = _x251 ∧ x5 = _x252 ∧ x6 = _x253 ∧ x7 = _x254 ∧ x8 = _x255 ∧ _x247 = _x255 ∧ _x246 = _x254 ∧ _x245 = _x253 ∧ _x244 = _x252 ∧ _x242 = _x250 ∧ _x243 = _x251 ∧ _x240 = _x248 ∧ _x241 = _x249

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 l4 l4 l4: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 l7 l7 l7: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 l10 l10 l10: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 l6 l6 l6: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 l11 l11 l11: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 l8 l8 l8: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 l1 l1 l1: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 l3 l3 l3: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 l0 l0 l0: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 l12 l12 l12: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 l2 l2 l2: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 l9 l9 l9: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8
and for every transition t, a duplicate t is considered.

### 2 SCC Decomposition

We consider subproblems for each of the 3 SCC(s) of the program graph.

### 2.1 SCC Subproblem 1/3

Here we consider the SCC { l1, l0 }.

### 2.1.1 Transition Removal

We remove transition 15 using the following ranking functions, which are bounded by 0.

 l0: 2⋅x2 − 2⋅x4 + 1 l1: 2⋅x2 − 2⋅x4

### 2.1.2 Transition Removal

We remove transition 1 using the following ranking functions, which are bounded by 0.

 l0: 0 l1: −1

### 2.1.3 Trivial Cooperation Program

There are no more "sharp" transitions in the cooperation program. Hence the cooperation termination is proved.

### 2.2 SCC Subproblem 2/3

Here we consider the SCC { l7, l10, l8, l3, l2, l9 }.

### 2.2.1 Transition Removal

We remove transition 13 using the following ranking functions, which are bounded by 0.

 l2: 4⋅x1 − 4⋅x5 + 1 l3: 4⋅x1 − 4⋅x5 l8: 4⋅x1 − 4⋅x5 − 1 l7: 4⋅x1 − 4⋅x5 − 1 l9: 4⋅x1 − 4⋅x5 − 1 l10: 4⋅x1 − 4⋅x5 − 1

### 2.2.2 Transition Removal

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

 l2: −1 l3: −2 l8: 0 l7: 0 l9: 0 l10: 0

### 2.2.3 Transition Removal

We remove transitions 2, 10 using the following ranking functions, which are bounded by 0.

 l2: 1 l3: 0 l7: −1 + x1 − x3 l8: −1 + x1 − x3 l9: −2 + x1 − x3 l10: −2 + x1 − x3

### 2.2.4 Transition Removal

We remove transitions 5, 6, 8, 7 using the following ranking functions, which are bounded by −2.

 l7: −2 l8: −3 l9: −1 l10: 0

### 2.2.5 Trivial Cooperation Program

There are no more "sharp" transitions in the cooperation program. Hence the cooperation termination is proved.

### 2.3 SCC Subproblem 3/3

Here we consider the SCC { l4, l6 }.

### 2.3.1 Transition Removal

We remove transition 4 using the following ranking functions, which are bounded by 0.

 l6: 2⋅x1 − 2⋅x5 + 1 l4: 2⋅x1 − 2⋅x5

### 2.3.2 Transition Removal

We remove transition 11 using the following ranking functions, which are bounded by 0.

 l6: 0 l4: −1

### 2.3.3 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 % (14 real / 0 unknown / 0 assumptions / 14 total proof steps)