# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: f795_0_main_GE, f1421_0_sort_GE, f1445_0_aux_LT, f1_0_main_Load, f1611_0_aux_InvokeMethod, __init
• Transitions: (pre-variables and post-variables)  f1_0_main_Load 1 f795_0_main_GE: x1 = _arg1 ∧ x2 = _arg2 ∧ x3 = _arg3 ∧ x4 = _arg4 ∧ x5 = _arg5 ∧ x6 = _arg6 ∧ x7 = _arg7 ∧ x8 = _arg8 ∧ x1 = _arg1P ∧ x2 = _arg2P ∧ x3 = _arg3P ∧ x4 = _arg4P ∧ x5 = _arg5P ∧ x6 = _arg6P ∧ x7 = _arg7P ∧ x8 = _arg8P ∧ 0 = _arg8P ∧ 0 = _arg7P ∧ 0 = _arg6P ∧ 0 = _arg5P ∧ 0 = _arg4P ∧ 0 = _arg3P ∧ 0 = _arg2 ∧ 0 ≤ _arg2P − 1 ∧ 0 ≤ _arg1P − 1 ∧ 0 ≤ _arg1 − 1 ∧ _arg2P ≤ _arg1 ∧ _arg1P ≤ _arg1 f1_0_main_Load 2 f795_0_main_GE: x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x5 = _x4 ∧ x6 = _x5 ∧ x7 = _x7 ∧ x8 = _x8 ∧ x1 = _x9 ∧ x2 = _x10 ∧ x3 = _x13 ∧ x4 = _x14 ∧ x5 = _x15 ∧ x6 = _x16 ∧ x7 = _x17 ∧ x8 = _x18 ∧ _x9 ≤ _x ∧ −1 ≤ _x19 − 1 ∧ _x10 ≤ _x ∧ 0 ≤ _x − 1 ∧ 0 ≤ _x9 − 1 ∧ 0 ≤ _x10 − 1 ∧ 1 = _x1 ∧ 0 = _x13 ∧ 0 = _x14 ∧ 1 = _x15 ∧ 1 = _x16 ∧ 1 = _x17 ∧ 0 = _x18 f1_0_main_Load 3 f795_0_main_GE: x1 = _x20 ∧ x2 = _x21 ∧ x3 = _x22 ∧ x4 = _x23 ∧ x5 = _x24 ∧ x6 = _x25 ∧ x7 = _x26 ∧ x8 = _x27 ∧ x1 = _x28 ∧ x2 = _x29 ∧ x3 = _x30 ∧ x4 = _x31 ∧ x5 = _x32 ∧ x6 = _x33 ∧ x7 = _x34 ∧ x8 = _x35 ∧ −1 ≤ _x36 − 1 ∧ 1 ≤ _x21 − 1 ∧ 0 ≤ _x36⋅_x37 ∧ −1 ≤ _x37 − 1 ∧ _x28 ≤ _x20 ∧ _x29 ≤ _x20 ∧ 0 ≤ _x20 − 1 ∧ 0 ≤ _x28 − 1 ∧ 0 ≤ _x29 − 1 ∧ 0 = _x30 ∧ _x36⋅_x37 = _x31 ∧ _x21 = _x32 ∧ 2 = _x33 ∧ _x21 = _x34 ∧ _x36⋅_x37 = _x35 f795_0_main_GE 4 f795_0_main_GE: x1 = _x38 ∧ x2 = _x39 ∧ x3 = _x40 ∧ x4 = _x41 ∧ x5 = _x42 ∧ x6 = _x43 ∧ x7 = _x44 ∧ x8 = _x45 ∧ x1 = _x46 ∧ x2 = _x47 ∧ x3 = _x48 ∧ x4 = _x49 ∧ x5 = _x50 ∧ x6 = _x51 ∧ x7 = _x52 ∧ x8 = _x53 ∧ _x41 = _x53 ∧ _x42 = _x52 ∧ _x43 = _x51 ∧ _x42 = _x50 ∧ _x41 = _x49 ∧ _x40 + 1 = _x48 ∧ _x41 = _x45 ∧ _x42 = _x44 ∧ 0 ≤ _x47 − 1 ∧ 0 ≤ _x46 − 1 ∧ 0 ≤ _x39 − 1 ∧ 0 ≤ _x38 − 1 ∧ _x47 ≤ _x39 ∧ _x47 ≤ _x38 ∧ _x46 ≤ _x39 ∧ _x46 ≤ _x38 ∧ −1 ≤ _x41 − 1 ∧ _x42 ≤ _x43 ∧ −1 ≤ _x42 − 1 ∧ _x40 ≤ _x41 − 1 f795_0_main_GE 5 f795_0_main_GE: x1 = _x54 ∧ x2 = _x55 ∧ x3 = _x56 ∧ x4 = _x57 ∧ x5 = _x58 ∧ x6 = _x59 ∧ x7 = _x60 ∧ x8 = _x61 ∧ x1 = _x62 ∧ x2 = _x63 ∧ x3 = _x64 ∧ x4 = _x65 ∧ x5 = _x66 ∧ x6 = _x67 ∧ x7 = _x70 ∧ x8 = _x71 ∧ _x57 = _x71 ∧ _x58 = _x70 ∧ _x59 + 1 = _x67 ∧ _x58 = _x66 ∧ _x57 = _x65 ∧ _x56 + 1 = _x64 ∧ _x57 = _x61 ∧ _x58 = _x60 ∧ 0 ≤ _x63 − 1 ∧ 0 ≤ _x62 − 1 ∧ 0 ≤ _x55 − 1 ∧ 0 ≤ _x54 − 1 ∧ _x63 ≤ _x55 ∧ _x63 ≤ _x54 ∧ _x62 ≤ _x55 ∧ _x62 ≤ _x54 ∧ −1 ≤ _x57 − 1 ∧ _x59 ≤ _x58 − 1 ∧ −1 ≤ _x59 − 1 ∧ −1 ≤ _x58 − 1 ∧ _x56 ≤ _x57 − 1 f795_0_main_GE 6 f1421_0_sort_GE: x1 = _x72 ∧ x2 = _x73 ∧ x3 = _x74 ∧ x4 = _x75 ∧ x5 = _x76 ∧ x6 = _x79 ∧ x7 = _x80 ∧ x8 = _x81 ∧ x1 = _x82 ∧ x2 = _x83 ∧ x3 = _x84 ∧ x4 = _x85 ∧ x5 = _x86 ∧ x6 = _x87 ∧ x7 = _x88 ∧ x8 = _x89 ∧ _x75 = _x86 ∧ _x75 = _x85 ∧ 1 = _x84 ∧ _x75 = _x81 ∧ _x76 = _x80 ∧ 0 ≤ _x83 − 1 ∧ 0 ≤ _x82 − 1 ∧ 0 ≤ _x73 − 1 ∧ 0 ≤ _x72 − 1 ∧ _x83 ≤ _x73 ∧ _x83 ≤ _x72 ∧ _x82 ≤ _x73 ∧ _x82 ≤ _x72 ∧ _x75 ≤ _x74 ∧ −1 ≤ _x75 − 1 f1421_0_sort_GE 7 f1421_0_sort_GE: x1 = _x90 ∧ x2 = _x91 ∧ x3 = _x92 ∧ x4 = _x93 ∧ x5 = _x94 ∧ x6 = _x95 ∧ x7 = _x96 ∧ x8 = _x97 ∧ x1 = _x98 ∧ x2 = _x99 ∧ x3 = _x100 ∧ x4 = _x101 ∧ x5 = _x102 ∧ x6 = _x103 ∧ x7 = _x104 ∧ x8 = _x105 ∧ _x94 = _x102 ∧ _x93 = _x101 ∧ _x92 + 1 = _x100 ∧ 0 ≤ _x99 − 1 ∧ 0 ≤ _x98 − 1 ∧ 0 ≤ _x91 − 1 ∧ 0 ≤ _x90 − 1 ∧ _x99 ≤ _x91 ∧ _x99 ≤ _x90 ∧ _x98 ≤ _x91 ∧ _x98 ≤ _x90 ∧ _x92 ≤ _x93 − 1 ∧ 0 ≤ _x92 − 1 ∧ 1 ≤ _x93 − 1 ∧ 1 ≤ _x93 − _x92 f1421_0_sort_GE 8 f1421_0_sort_GE: x1 = _x106 ∧ x2 = _x107 ∧ x3 = _x108 ∧ x4 = _x109 ∧ x5 = _x110 ∧ x6 = _x111 ∧ x7 = _x112 ∧ x8 = _x113 ∧ x1 = _x114 ∧ x2 = _x115 ∧ x3 = _x116 ∧ x4 = _x117 ∧ x5 = _x118 ∧ x6 = _x119 ∧ x7 = _x120 ∧ x8 = _x121 ∧ _x109 = _x117 ∧ _x108 + 1 = _x116 ∧ 0 ≤ _x115 − 1 ∧ 0 ≤ _x114 − 1 ∧ 0 ≤ _x107 − 1 ∧ 0 ≤ _x106 − 1 ∧ _x115 ≤ _x107 ∧ _x115 ≤ _x106 ∧ _x114 ≤ _x107 ∧ _x114 ≤ _x106 ∧ _x108 ≤ _x109 − 1 ∧ 0 ≤ _x108 − 1 ∧ 1 ≤ _x109 − 1 ∧ 1 ≤ _x109 − _x108 f1421_0_sort_GE 9 f1445_0_aux_LT: x1 = _x122 ∧ x2 = _x123 ∧ x3 = _x124 ∧ x4 = _x125 ∧ x5 = _x126 ∧ x6 = _x127 ∧ x7 = _x128 ∧ x8 = _x129 ∧ x1 = _x130 ∧ x2 = _x131 ∧ x3 = _x132 ∧ x4 = _x133 ∧ x5 = _x134 ∧ x6 = _x135 ∧ x7 = _x136 ∧ x8 = _x137 ∧ _x126 = _x135 ∧ _x125 − _x124 = _x134 ∧ 0 = _x133 ∧ 0 = _x131 ∧ 0 = _x130 ∧ 0 ≤ _x132 − 1 ∧ 0 ≤ _x123 − 1 ∧ 0 ≤ _x122 − 1 ∧ _x132 ≤ _x123 ∧ _x132 ≤ _x122 ∧ _x124 ≤ _x125 − 1 ∧ 0 ≤ _x124 − 1 ∧ 1 ≤ _x125 − 1 ∧ 1 ≤ _x125 − _x124 f1445_0_aux_LT 10 f1611_0_aux_InvokeMethod: x1 = _x138 ∧ x2 = _x139 ∧ x3 = _x140 ∧ x4 = _x141 ∧ x5 = _x142 ∧ x6 = _x143 ∧ x7 = _x144 ∧ x8 = _x145 ∧ x1 = _x146 ∧ x2 = _x147 ∧ x3 = _x148 ∧ x4 = _x149 ∧ x5 = _x150 ∧ x6 = _x151 ∧ x7 = _x152 ∧ x8 = _x153 ∧ −1 ≤ _x139 − 1 ∧ _x139 + 1 ≤ _x143 − 1 ∧ _x139 ≤ _x142 − 1 ∧ _x154 ≤ _x155 ∧ _x149 ≤ _x140 ∧ 0 ≤ _x140 − 1 ∧ 0 ≤ _x149 − 1 ∧ _x139 = _x141 ∧ _x138 = _x146 ∧ _x139 + 1 = _x147 ∧ _x142 = _x148 ∧ _x143 = _x151 f1445_0_aux_LT 11 f1611_0_aux_InvokeMethod: x1 = _x156 ∧ x2 = _x157 ∧ x3 = _x158 ∧ x4 = _x159 ∧ x5 = _x160 ∧ x6 = _x161 ∧ x7 = _x162 ∧ x8 = _x163 ∧ x1 = _x164 ∧ x2 = _x165 ∧ x3 = _x166 ∧ x4 = _x167 ∧ x5 = _x168 ∧ x6 = _x169 ∧ x7 = _x170 ∧ x8 = _x171 ∧ −1 ≤ _x157 − 1 ∧ _x157 + 1 ≤ _x161 − 1 ∧ _x157 ≤ _x160 − 1 ∧ _x172 ≤ _x173 − 1 ∧ _x167 ≤ _x158 ∧ 0 ≤ _x158 − 1 ∧ 0 ≤ _x167 − 1 ∧ _x157 = _x159 ∧ _x156 = _x164 ∧ _x157 + 1 = _x165 ∧ _x160 = _x166 ∧ _x161 = _x169 f1611_0_aux_InvokeMethod 12 f1445_0_aux_LT: x1 = _x174 ∧ x2 = _x175 ∧ x3 = _x176 ∧ x4 = _x177 ∧ x5 = _x178 ∧ x6 = _x179 ∧ x7 = _x180 ∧ x8 = _x181 ∧ x1 = _x182 ∧ x2 = _x183 ∧ x3 = _x184 ∧ x4 = _x185 ∧ x5 = _x186 ∧ x6 = _x187 ∧ x7 = _x188 ∧ x8 = _x189 ∧ _x179 = _x187 ∧ _x176 = _x186 ∧ _x175 = _x185 ∧ _x175 = _x183 ∧ _x175 = _x182 ∧ 0 ≤ _x184 − 1 ∧ 0 ≤ _x177 − 1 ∧ _x184 ≤ _x177 ∧ _x174 ≤ _x176 ∧ _x175 ≤ _x176 ∧ 0 ≤ _x175 − 1 ∧ 0 ≤ _x176 − 1 __init 13 f1_0_main_Load: x1 = _x190 ∧ x2 = _x191 ∧ x3 = _x192 ∧ x4 = _x193 ∧ x5 = _x194 ∧ x6 = _x195 ∧ x7 = _x196 ∧ x8 = _x197 ∧ x1 = _x198 ∧ x2 = _x199 ∧ x3 = _x200 ∧ x4 = _x201 ∧ x5 = _x202 ∧ x6 = _x203 ∧ x7 = _x204 ∧ x8 = _x205 ∧ 0 ≤ 0

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 f795_0_main_GE f795_0_main_GE f795_0_main_GE: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 f1421_0_sort_GE f1421_0_sort_GE f1421_0_sort_GE: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 f1445_0_aux_LT f1445_0_aux_LT f1445_0_aux_LT: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 f1_0_main_Load f1_0_main_Load f1_0_main_Load: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 f1611_0_aux_InvokeMethod f1611_0_aux_InvokeMethod f1611_0_aux_InvokeMethod: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 __init __init __init: 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 { f795_0_main_GE }.

### 2.1.1 Transition Removal

We remove transitions 4, 5 using the following ranking functions, which are bounded by 0.

 f795_0_main_GE: − x3 + x4

### 2.1.2 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 { f1421_0_sort_GE }.

### 2.2.1 Transition Removal

We remove transition 8 using the following ranking functions, which are bounded by 0.

 f1421_0_sort_GE: −1 − x3 + x4

### 2.2.2 Transition Removal

We remove transition 7 using the following ranking functions, which are bounded by 0.

 f1421_0_sort_GE: − x3 + x4

### 2.2.3 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 { f1445_0_aux_LT, f1611_0_aux_InvokeMethod }.

### 2.3.1 Transition Removal

We remove transitions 10, 12, 11 using the following ranking functions, which are bounded by 0.

 f1445_0_aux_LT: −1 − 2⋅x2 + 2⋅x5 f1611_0_aux_InvokeMethod: −2⋅x2 + 2⋅x3

### 2.3.2 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 % (11 real / 0 unknown / 0 assumptions / 11 total proof steps)