# 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 = _acc12HAT0 ∧ x2 = _acc_length11HAT0 ∧ x3 = _coef_len210HAT0 ∧ x4 = _coef_len6HAT0 ∧ x5 = _i8HAT0 ∧ x6 = _in_len4HAT0 ∧ x7 = _j9HAT0 ∧ x8 = _scale7HAT0 ∧ x1 = _acc12HATpost ∧ x2 = _acc_length11HATpost ∧ x3 = _coef_len210HATpost ∧ x4 = _coef_len6HATpost ∧ x5 = _i8HATpost ∧ x6 = _in_len4HATpost ∧ x7 = _j9HATpost ∧ x8 = _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 l2 2 l0: 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 ∧ _x3 = _x11 ∧ _x2 = _x10 ∧ _x1 = _x9 ∧ _x = _x8 ∧ _x3 ≤ _x1 l2 3 l0: 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 ∧ _x19 = _x27 ∧ _x18 = _x26 ∧ _x16 = _x24 ∧ _x25 = 1 + _x17 ∧ 1 + _x17 ≤ _x19 l1 4 l3: 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 ∧ _x35 = _x43 ∧ _x34 = _x42 ∧ _x33 = _x41 ∧ _x32 = _x40 ∧ _x44 = 1 + _x36 l4 5 l2: 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 ∧ _x51 = _x59 ∧ _x50 = _x58 ∧ _x49 = _x57 ∧ _x48 = _x56 l4 6 l1: 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 ∧ _x66 = _x74 ∧ _x64 = _x72 ∧ _x73 = −1 + _x65 l3 7 l5: 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 ∧ _x85 = _x93 ∧ _x84 = _x92 ∧ _x83 = _x91 ∧ _x82 = _x90 ∧ _x81 = _x89 ∧ _x80 = _x88 l6 8 l4: 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 ∧ _x99 = _x107 ∧ _x98 = _x106 ∧ _x97 = _x105 ∧ _x96 = _x104 ∧ _x97 ≤ _x102 l6 9 l7: 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 ∧ _x116 = _x124 ∧ _x115 = _x123 ∧ _x114 = _x122 ∧ _x113 = _x121 ∧ _x126 = 1 + _x118 ∧ _x120 = _x120 ∧ 1 + _x118 ≤ _x113 l7 10 l6: 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 ∧ _x131 = _x139 ∧ _x130 = _x138 ∧ _x129 = _x137 ∧ _x128 = _x136 l5 11 l8: 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 ∧ _x147 = _x155 ∧ _x146 = _x154 ∧ _x145 = _x153 ∧ _x144 = _x152 ∧ _x149 ≤ _x148 l5 12 l7: 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 ∧ _x165 = _x173 ∧ _x164 = _x172 ∧ _x163 = _x171 ∧ _x162 = _x170 ∧ _x161 = _x169 ∧ _x174 = 1 ∧ _x168 = _x168 ∧ 1 + _x164 ≤ _x165 l9 13 l3: 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 ∧ _x182 = _x190 ∧ _x176 = _x184 ∧ _x188 = 0 ∧ _x185 = _x186 ∧ _x186 = _x186 ∧ _x191 = 285 ∧ _x187 = 35 ∧ _x189 = 10 l10 14 l9: 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 ∧ _x196 = _x204 ∧ _x195 = _x203 ∧ _x194 = _x202 ∧ _x193 = _x201 ∧ _x192 = _x200

## 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 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 l6 l6 l6: 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 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 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 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 − x5 + x6 l1: −2 − x5 + x6 l2: −2 − x5 + x6 l4: −2 − x5 + x6 l6: −2 − x5 + x6 l7: −2 − x5 + x6 l5: −1 − x5 + x6 l3: −1 − x5 + x6

### 2.1.2 Transition Removal

We remove transition 3 using the following ranking functions, which are bounded by 0.

 l0: −2 − x2 + x4 l1: −2 − x2 + x4 l2: −1 − x2 + x4 l4: −1 − x2 + x4 l6: −1 − x2 + x4 l7: −1 − x2 + x4 l3: −2 − x2 + x4 l5: −2 − x2 + x4

### 2.1.3 Transition Removal

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

 l0: 3 l1: 2 l2: 4 l4: 5 l6: 6 l7: 6 l3: 1 l5: 0

### 2.1.4 Transition Removal

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

 l7: 2⋅x2 − 2⋅x7 + 1 l6: 2⋅x2 − 2⋅x7

### 2.1.5 Transition Removal

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

 l7: 0 l6: −1

### 2.1.6 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 % (9 real / 0 unknown / 0 assumptions / 9 total proof steps)