LTS Termination Proof

by AProVE

Input

Integer Transition System
• Initial Location: f480_0_main_InvokeMethod, f177_0_appendNewList_Return, f415_0_main_InvokeMethod, f1_0_main_Load, f519_0_length_NONNULL, __init, f343_0_appendNewList_GT
• Transitions: (pre-variables and post-variables)  f1_0_main_Load 1 f177_0_appendNewList_Return: x1 = _arg1 ∧ x2 = _arg2 ∧ x3 = _arg3 ∧ x4 = _arg4 ∧ x1 = _arg1P ∧ x2 = _arg2P ∧ x3 = _arg3P ∧ x4 = _arg4P ∧ −1 ≤ _x5 − 1 ∧ 0 ≤ _arg2 − 1 ∧ _arg1P ≤ _arg1 ∧ _arg2P − 7 ≤ _arg1 ∧ _arg3P − 5 ≤ _arg1 ∧ 0 ≤ _arg1 − 1 ∧ 0 ≤ _arg1P − 1 ∧ 7 ≤ _arg2P − 1 ∧ 5 ≤ _arg3P − 1 ∧ 0 = _arg4P f1_0_main_Load 2 f177_0_appendNewList_Return: x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x1 = _x4 ∧ x2 = _x6 ∧ x3 = _x7 ∧ x4 = _x8 ∧ −1 ≤ _x9 − 1 ∧ 0 ≤ _x1 − 1 ∧ _x4 ≤ _x ∧ 0 ≤ _x − 1 ∧ 0 ≤ _x4 − 1 ∧ 8 ≤ _x6 − 1 ∧ 6 ≤ _x7 − 1 f177_0_appendNewList_Return 3 f415_0_main_InvokeMethod: x1 = _x10 ∧ x2 = _x11 ∧ x3 = _x14 ∧ x4 = _x15 ∧ x1 = _x16 ∧ x2 = _x17 ∧ x3 = _x18 ∧ x4 = _x21 ∧ −1 ≤ _x22 − 1 ∧ 1 ≤ _x23 − 1 ∧ _x16 ≤ _x10 ∧ _x16 + 7 ≤ _x11 ∧ _x16 + 5 ≤ _x14 ∧ 0 ≤ _x10 − 1 ∧ 7 ≤ _x11 − 1 ∧ 5 ≤ _x14 − 1 ∧ 0 ≤ _x16 − 1 f177_0_appendNewList_Return 4 f415_0_main_InvokeMethod: x1 = _x24 ∧ x2 = _x25 ∧ x3 = _x30 ∧ x4 = _x31 ∧ x1 = _x32 ∧ x2 = _x36 ∧ x3 = _x37 ∧ x4 = _x38 ∧ _x31 ≤ _x39 − 1 ∧ 0 ≤ _x31 − 1 ∧ _x31 ≤ _x40 − 1 ∧ _x31 ≤ _x41 − 1 ∧ 1 ≤ _x47 − 1 ∧ −1 ≤ _x48 − 1 ∧ _x32 ≤ _x24 ∧ _x32 + 6 ≤ _x25 ∧ _x32 + 4 ≤ _x30 ∧ 0 ≤ _x24 − 1 ∧ 6 ≤ _x25 − 1 ∧ 4 ≤ _x30 − 1 ∧ 0 ≤ _x32 − 1 f1_0_main_Load 5 f415_0_main_InvokeMethod: x1 = _x49 ∧ x2 = _x50 ∧ x3 = _x51 ∧ x4 = _x52 ∧ x1 = _x55 ∧ x2 = _x56 ∧ x3 = _x57 ∧ x4 = _x58 ∧ −1 ≤ _x59 − 1 ∧ 1 ≤ _x50 − 1 ∧ −1 ≤ _x60 − 1 ∧ _x55 ≤ _x49 ∧ 0 ≤ _x49 − 1 ∧ 0 ≤ _x55 − 1 f177_0_appendNewList_Return 6 f480_0_main_InvokeMethod: x1 = _x62 ∧ x2 = _x63 ∧ x3 = _x64 ∧ x4 = _x65 ∧ x1 = _x69 ∧ x2 = _x70 ∧ x3 = _x71 ∧ x4 = _x72 ∧ _x65 ≤ _x73 − 1 ∧ 0 ≤ _x65 − 1 ∧ _x65 ≤ _x74 − 1 ∧ _x65 ≤ _x79 − 1 ∧ 1 ≤ _x80 − 1 ∧ −1 ≤ _x81 − 1 ∧ _x69 ≤ _x62 ∧ _x69 + 6 ≤ _x63 ∧ _x69 + 4 ≤ _x64 ∧ 0 ≤ _x62 − 1 ∧ 6 ≤ _x63 − 1 ∧ 4 ≤ _x64 − 1 ∧ 0 ≤ _x69 − 1 ∧ 6 ≤ _x70 − 1 f177_0_appendNewList_Return 7 f480_0_main_InvokeMethod: x1 = _x82 ∧ x2 = _x85 ∧ x3 = _x86 ∧ x4 = _x87 ∧ x1 = _x89 ∧ x2 = _x90 ∧ x3 = _x91 ∧ x4 = _x92 ∧ −1 ≤ _x93 − 1 ∧ 1 ≤ _x98 − 1 ∧ _x89 ≤ _x82 ∧ _x89 + 7 ≤ _x85 ∧ _x89 + 5 ≤ _x86 ∧ _x90 − 5 ≤ _x82 ∧ _x90 + 2 ≤ _x85 ∧ _x90 ≤ _x86 ∧ 0 ≤ _x82 − 1 ∧ 7 ≤ _x85 − 1 ∧ 5 ≤ _x86 − 1 ∧ 0 ≤ _x89 − 1 ∧ 5 ≤ _x90 − 1 f177_0_appendNewList_Return 8 f480_0_main_InvokeMethod: x1 = _x99 ∧ x2 = _x100 ∧ x3 = _x101 ∧ x4 = _x102 ∧ x1 = _x103 ∧ x2 = _x104 ∧ x3 = _x105 ∧ x4 = _x107 ∧ −1 ≤ _x108 − 1 ∧ 1 ≤ _x109 − 1 ∧ _x103 ≤ _x99 ∧ _x103 + 7 ≤ _x100 ∧ _x103 + 5 ≤ _x101 ∧ 0 ≤ _x99 − 1 ∧ 7 ≤ _x100 − 1 ∧ 5 ≤ _x101 − 1 ∧ 0 ≤ _x103 − 1 ∧ 6 ≤ _x104 − 1 f1_0_main_Load 9 f480_0_main_InvokeMethod: x1 = _x111 ∧ x2 = _x112 ∧ x3 = _x113 ∧ x4 = _x115 ∧ x1 = _x116 ∧ x2 = _x117 ∧ x3 = _x118 ∧ x4 = _x119 ∧ −1 ≤ _x120 − 1 ∧ 1 ≤ _x112 − 1 ∧ −1 ≤ _x121 − 1 ∧ _x116 ≤ _x111 ∧ 0 ≤ _x111 − 1 ∧ 0 ≤ _x116 − 1 ∧ 6 ≤ _x117 − 1 f177_0_appendNewList_Return 10 f480_0_main_InvokeMethod: x1 = _x122 ∧ x2 = _x123 ∧ x3 = _x124 ∧ x4 = _x125 ∧ x1 = _x126 ∧ x2 = _x127 ∧ x3 = _x128 ∧ x4 = _x129 ∧ _x125 ≤ _x130 − 1 ∧ 0 ≤ _x125 − 1 ∧ _x125 ≤ _x131 − 1 ∧ _x125 ≤ _x132 − 1 ∧ 1 ≤ _x133 − 1 ∧ −1 ≤ _x134 − 1 ∧ _x126 ≤ _x122 ∧ _x126 + 6 ≤ _x123 ∧ _x126 + 4 ≤ _x124 ∧ _x127 − 5 ≤ _x122 ∧ _x127 + 1 ≤ _x123 ∧ _x127 − 1 ≤ _x124 ∧ 0 ≤ _x122 − 1 ∧ 6 ≤ _x123 − 1 ∧ 4 ≤ _x124 − 1 ∧ 0 ≤ _x126 − 1 ∧ 5 ≤ _x127 − 1 f1_0_main_Load 11 f480_0_main_InvokeMethod: x1 = _x135 ∧ x2 = _x136 ∧ x3 = _x137 ∧ x4 = _x138 ∧ x1 = _x139 ∧ x2 = _x140 ∧ x3 = _x141 ∧ x4 = _x142 ∧ −1 ≤ _x143 − 1 ∧ 1 ≤ _x136 − 1 ∧ −1 ≤ _x144 − 1 ∧ _x139 ≤ _x135 ∧ _x140 − 5 ≤ _x135 ∧ 0 ≤ _x135 − 1 ∧ 0 ≤ _x139 − 1 ∧ 5 ≤ _x140 − 1 f1_0_main_Load 12 f343_0_appendNewList_GT: x1 = _x145 ∧ x2 = _x146 ∧ x3 = _x147 ∧ x4 = _x148 ∧ x1 = _x149 ∧ x2 = _x150 ∧ x3 = _x151 ∧ x4 = _x152 ∧ 1 = _x150 ∧ 0 ≤ _x145 − 1 ∧ 0 ≤ _x146 − 1 ∧ −1 ≤ _x149 − 1 f177_0_appendNewList_Return 13 f343_0_appendNewList_GT: x1 = _x153 ∧ x2 = _x154 ∧ x3 = _x155 ∧ x4 = _x156 ∧ x1 = _x157 ∧ x2 = _x158 ∧ x3 = _x159 ∧ x4 = _x160 ∧ _x156 ≤ _x161 − 1 ∧ 0 ≤ _x156 − 1 ∧ _x156 ≤ _x162 − 1 ∧ _x156 ≤ _x163 − 1 ∧ 1 ≤ _x164 − 1 ∧ −1 ≤ _x157 − 1 ∧ 0 ≤ _x153 − 1 ∧ 6 ≤ _x154 − 1 ∧ 4 ≤ _x155 − 1 ∧ 2 = _x158 f177_0_appendNewList_Return 14 f343_0_appendNewList_GT: x1 = _x165 ∧ x2 = _x166 ∧ x3 = _x167 ∧ x4 = _x168 ∧ x1 = _x169 ∧ x2 = _x170 ∧ x3 = _x171 ∧ x4 = _x172 ∧ −1 ≤ _x169 − 1 ∧ 1 ≤ _x173 − 1 ∧ 0 ≤ _x165 − 1 ∧ 7 ≤ _x166 − 1 ∧ 5 ≤ _x167 − 1 ∧ 2 = _x170 f1_0_main_Load 15 f343_0_appendNewList_GT: x1 = _x174 ∧ x2 = _x175 ∧ x3 = _x176 ∧ x4 = _x177 ∧ x1 = _x178 ∧ x2 = _x179 ∧ x3 = _x180 ∧ x4 = _x181 ∧ −1 ≤ _x182 − 1 ∧ 1 ≤ _x175 − 1 ∧ −1 ≤ _x178 − 1 ∧ 0 ≤ _x174 − 1 ∧ 2 = _x179 f343_0_appendNewList_GT 16 f343_0_appendNewList_GT: x1 = _x183 ∧ x2 = _x184 ∧ x3 = _x185 ∧ x4 = _x186 ∧ x1 = _x187 ∧ x2 = _x188 ∧ x3 = _x189 ∧ x4 = _x190 ∧ _x184 = _x188 ∧ _x183 − 1 = _x187 ∧ 1 ≤ _x183 − 1 ∧ 0 ≤ _x184 − 1 ∧ _x183 − 1 ≤ _x183 − 1 f415_0_main_InvokeMethod 17 f519_0_length_NONNULL: x1 = _x191 ∧ x2 = _x192 ∧ x3 = _x193 ∧ x4 = _x194 ∧ x1 = _x195 ∧ x2 = _x196 ∧ x3 = _x197 ∧ x4 = _x198 ∧ _x195 − 3 ≤ _x191 ∧ 1 ≤ _x199 − 1 ∧ _x196 − 1 ≤ _x191 ∧ 0 ≤ _x191 − 1 ∧ 3 ≤ _x195 − 1 ∧ 1 ≤ _x196 − 1 f480_0_main_InvokeMethod 18 f519_0_length_NONNULL: x1 = _x200 ∧ x2 = _x201 ∧ x3 = _x202 ∧ x4 = _x203 ∧ x1 = _x204 ∧ x2 = _x205 ∧ x3 = _x206 ∧ x4 = _x207 ∧ _x204 ≤ _x201 ∧ 1 ≤ _x208 − 1 ∧ _x205 + 2 ≤ _x201 ∧ 0 ≤ _x200 − 1 ∧ 4 ≤ _x201 − 1 ∧ 4 ≤ _x204 − 1 ∧ 2 ≤ _x205 − 1 f519_0_length_NONNULL 19 f519_0_length_NONNULL: x1 = _x209 ∧ x2 = _x210 ∧ x3 = _x211 ∧ x4 = _x212 ∧ x1 = _x213 ∧ x2 = _x214 ∧ x3 = _x215 ∧ x4 = _x216 ∧ −1 ≤ _x214 − 1 ∧ 0 ≤ _x213 − 1 ∧ 0 ≤ _x210 − 1 ∧ 2 ≤ _x209 − 1 ∧ _x214 + 1 ≤ _x210 ∧ _x214 + 3 ≤ _x209 ∧ _x213 ≤ _x210 ∧ _x213 + 2 ≤ _x209 __init 20 f1_0_main_Load: x1 = _x217 ∧ x2 = _x218 ∧ x3 = _x219 ∧ x4 = _x220 ∧ x1 = _x221 ∧ x2 = _x222 ∧ x3 = _x223 ∧ x4 = _x224 ∧ 0 ≤ 0

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 f480_0_main_InvokeMethod f480_0_main_InvokeMethod f480_0_main_InvokeMethod: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 f177_0_appendNewList_Return f177_0_appendNewList_Return f177_0_appendNewList_Return: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 f415_0_main_InvokeMethod f415_0_main_InvokeMethod f415_0_main_InvokeMethod: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 f1_0_main_Load f1_0_main_Load f1_0_main_Load: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 f519_0_length_NONNULL f519_0_length_NONNULL f519_0_length_NONNULL: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 __init __init __init: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 f343_0_appendNewList_GT f343_0_appendNewList_GT f343_0_appendNewList_GT: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4
and for every transition t, a duplicate t is considered.

2 SCC Decomposition

We consider subproblems for each of the 2 SCC(s) of the program graph.

2.1 SCC Subproblem 1/2

Here we consider the SCC { f343_0_appendNewList_GT }.

2.1.1 Transition Removal

We remove transition 16 using the following ranking functions, which are bounded by 0.

 f343_0_appendNewList_GT: x1

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 2/2

Here we consider the SCC { f519_0_length_NONNULL }.

2.2.1 Transition Removal

We remove transition 19 using the following ranking functions, which are bounded by 0.

 f519_0_length_NONNULL: x2 + x1

2.2.2 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 % (8 real / 0 unknown / 0 assumptions / 8 total proof steps)