# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: f500_0_main_GE, f493_0_rec4_LT, f376_0_rec1_GT, f1_0_main_ConstantStackPush, f541_0_rec3_GT, f323_0_rec0_GT, f319_0_rec2_LT, __init
• Transitions: (pre-variables and post-variables)  f1_0_main_ConstantStackPush 1 f500_0_main_GE: x1 = _arg1 ∧ x2 = _arg2 ∧ x3 = _arg3 ∧ x4 = _arg4 ∧ x5 = _arg5 ∧ x1 = _arg1P ∧ x2 = _arg2P ∧ x3 = _arg3P ∧ x4 = _arg4P ∧ x5 = _arg5P ∧ _arg2 = _arg3P ∧ 0 = _arg2P ∧ 0 ≤ _arg1P − 1 ∧ 0 ≤ _arg1 − 1 ∧ −1 ≤ _arg2 − 1 ∧ _arg1P ≤ _arg1 f500_0_main_GE 2 f500_0_main_GE: x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x5 = _x4 ∧ x1 = _x5 ∧ x2 = _x6 ∧ x3 = _x7 ∧ x4 = _x8 ∧ x5 = _x9 ∧ _x2 = _x7 ∧ _x1 + 1 = _x6 ∧ 0 ≤ _x5 − 1 ∧ 0 ≤ _x − 1 ∧ _x5 ≤ _x ∧ 0 ≤ _x2 − 1 ∧ _x1 ≤ _x2 − 1 f500_0_main_GE 3 f323_0_rec0_GT: x1 = _x10 ∧ x2 = _x11 ∧ x3 = _x12 ∧ x4 = _x13 ∧ x5 = _x14 ∧ x1 = _x15 ∧ x2 = _x16 ∧ x3 = _x17 ∧ x4 = _x18 ∧ x5 = _x19 ∧ _x12 = _x16 ∧ 0 = _x15 ∧ 0 ≤ _x10 − 1 ∧ 0 ≤ _x12 − 1 ∧ _x11 ≤ _x12 − 1 f323_0_rec0_GT 4 f323_0_rec0_GT: x1 = _x20 ∧ x2 = _x21 ∧ x3 = _x22 ∧ x4 = _x23 ∧ x5 = _x24 ∧ x1 = _x25 ∧ x2 = _x26 ∧ x3 = _x27 ∧ x4 = _x28 ∧ x5 = _x29 ∧ _x21 = _x26 ∧ _x20 + 1 = _x25 ∧ 0 ≤ _x21 − 1 ∧ −1 ≤ _x20 − 1 ∧ _x20 ≤ _x21 f323_0_rec0_GT 5 f376_0_rec1_GT: x1 = _x30 ∧ x2 = _x31 ∧ x3 = _x32 ∧ x4 = _x33 ∧ x5 = _x34 ∧ x1 = _x35 ∧ x2 = _x36 ∧ x3 = _x37 ∧ x4 = _x38 ∧ x5 = _x39 ∧ 2⋅_x30 = _x37 ∧ 0 = _x36 ∧ _x30 = _x35 ∧ _x30 ≤ _x31 ∧ 0 ≤ _x31 − 1 f376_0_rec1_GT 6 f376_0_rec1_GT: x1 = _x40 ∧ x2 = _x41 ∧ x3 = _x42 ∧ x4 = _x43 ∧ x5 = _x44 ∧ x1 = _x45 ∧ x2 = _x46 ∧ x3 = _x47 ∧ x4 = _x48 ∧ x5 = _x49 ∧ _x42 = _x47 ∧ _x41 + 1 = _x46 ∧ _x40 = _x45 ∧ −1 ≤ _x41 − 1 ∧ _x41 ≤ _x42 ∧ −1 ≤ _x40 − 1 f376_0_rec1_GT 7 f319_0_rec2_LT: x1 = _x50 ∧ x2 = _x51 ∧ x3 = _x52 ∧ x4 = _x53 ∧ x5 = _x54 ∧ x1 = _x55 ∧ x2 = _x56 ∧ x3 = _x57 ∧ x4 = _x58 ∧ x5 = _x59 ∧ _x50 + _x51 = _x57 ∧ _x51 = _x56 ∧ _x50 = _x55 ∧ −1 ≤ _x51 − 1 ∧ _x51 ≤ _x52 ∧ −1 ≤ _x50 − 1 f319_0_rec2_LT 8 f319_0_rec2_LT: x1 = _x60 ∧ x2 = _x61 ∧ x3 = _x62 ∧ x4 = _x63 ∧ x5 = _x64 ∧ x1 = _x65 ∧ x2 = _x66 ∧ x3 = _x67 ∧ x4 = _x68 ∧ x5 = _x69 ∧ _x62 − 1 = _x67 ∧ _x61 = _x66 ∧ _x60 = _x65 ∧ 0 ≤ 4⋅_x62 ∧ 0 ≤ 2⋅_x60 + 3⋅_x61 ∧ 0 ≤ 2⋅_x60 ∧ −1 ≤ _x62 − 1 ∧ 0 ≤ 3⋅_x61 f319_0_rec2_LT 9 f541_0_rec3_GT: x1 = _x70 ∧ x2 = _x71 ∧ x3 = _x72 ∧ x4 = _x73 ∧ x5 = _x74 ∧ x1 = _x75 ∧ x2 = _x76 ∧ x3 = _x77 ∧ x4 = _x78 ∧ x5 = _x79 ∧ 2⋅_x70 + 3⋅_x71 + 4⋅_x72 = _x79 ∧ 0 = _x78 ∧ _x72 = _x77 ∧ _x71 = _x76 ∧ _x70 = _x75 ∧ 0 ≤ 4⋅_x72 ∧ 0 ≤ 2⋅_x70 + 3⋅_x71 ∧ 0 ≤ 2⋅_x70 ∧ −1 ≤ _x72 − 1 ∧ 0 ≤ 3⋅_x71 f541_0_rec3_GT 10 f541_0_rec3_GT: x1 = _x80 ∧ x2 = _x81 ∧ x3 = _x82 ∧ x4 = _x83 ∧ x5 = _x84 ∧ x1 = _x85 ∧ x2 = _x86 ∧ x3 = _x87 ∧ x4 = _x88 ∧ x5 = _x89 ∧ _x84 = _x89 ∧ _x83 + 1 = _x88 ∧ _x82 = _x87 ∧ _x81 = _x86 ∧ _x80 = _x85 ∧ −1 ≤ _x83 − 1 ∧ 0 ≤ 1000⋅_x80 + 100⋅_x81 + 10⋅_x82 ∧ −1 ≤ _x82 − 1 ∧ 0 ≤ 1000⋅_x80 + 100⋅_x81 ∧ 0 ≤ 10⋅_x82 ∧ 0 ≤ 1000⋅_x80 ∧ _x83 ≤ _x84 ∧ 0 ≤ 100⋅_x81 f541_0_rec3_GT 11 f493_0_rec4_LT: x1 = _x90 ∧ x2 = _x91 ∧ x3 = _x92 ∧ x4 = _x93 ∧ x5 = _x94 ∧ x1 = _x95 ∧ x2 = _x96 ∧ x3 = _x97 ∧ x4 = _x98 ∧ x5 = _x99 ∧ 1000⋅_x90 + 100⋅_x91 + 10⋅_x92 + _x93 = _x95 ∧ 0 ≤ 1000⋅_x90 + 100⋅_x91 + 10⋅_x92 ∧ −1 ≤ _x93 − 1 ∧ −1 ≤ _x92 − 1 ∧ 0 ≤ 1000⋅_x90 + 100⋅_x91 ∧ 0 ≤ 10⋅_x92 ∧ 0 ≤ 1000⋅_x90 ∧ _x93 ≤ _x94 ∧ 0 ≤ 100⋅_x91 f493_0_rec4_LT 12 f493_0_rec4_LT: x1 = _x100 ∧ x2 = _x101 ∧ x3 = _x102 ∧ x4 = _x103 ∧ x5 = _x104 ∧ x1 = _x105 ∧ x2 = _x106 ∧ x3 = _x107 ∧ x4 = _x108 ∧ x5 = _x109 ∧ _x100 − 1 = _x105 ∧ −1 ≤ _x100 − 1 __init 13 f1_0_main_ConstantStackPush: x1 = _x110 ∧ x2 = _x111 ∧ x3 = _x112 ∧ x4 = _x113 ∧ x5 = _x114 ∧ x1 = _x115 ∧ x2 = _x116 ∧ x3 = _x117 ∧ x4 = _x118 ∧ x5 = _x119 ∧ 0 ≤ 0

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 f500_0_main_GE f500_0_main_GE f500_0_main_GE: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 f493_0_rec4_LT f493_0_rec4_LT f493_0_rec4_LT: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 f376_0_rec1_GT f376_0_rec1_GT f376_0_rec1_GT: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 f1_0_main_ConstantStackPush f1_0_main_ConstantStackPush f1_0_main_ConstantStackPush: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 f541_0_rec3_GT f541_0_rec3_GT f541_0_rec3_GT: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 f323_0_rec0_GT f323_0_rec0_GT f323_0_rec0_GT: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 f319_0_rec2_LT f319_0_rec2_LT f319_0_rec2_LT: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 __init __init __init: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5
and for every transition t, a duplicate t is considered.

### 2 SCC Decomposition

We consider subproblems for each of the 6 SCC(s) of the program graph.

### 2.1 SCC Subproblem 1/6

Here we consider the SCC { f500_0_main_GE }.

### 2.1.1 Transition Removal

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

 f500_0_main_GE: x3 − x2

### 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 2/6

Here we consider the SCC { f323_0_rec0_GT }.

### 2.2.1 Transition Removal

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

 f323_0_rec0_GT: − x1 + x2

### 2.2.2 Trivial Cooperation Program

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

### 2.3 SCC Subproblem 3/6

Here we consider the SCC { f376_0_rec1_GT }.

### 2.3.1 Transition Removal

We remove transition 6 using the following ranking functions, which are bounded by 0.

 f376_0_rec1_GT: − x2 + x3

### 2.3.2 Trivial Cooperation Program

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

### 2.4 SCC Subproblem 4/6

Here we consider the SCC { f319_0_rec2_LT }.

### 2.4.1 Transition Removal

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

 f319_0_rec2_LT: x3

### 2.4.2 Trivial Cooperation Program

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

### 2.5 SCC Subproblem 5/6

Here we consider the SCC { f541_0_rec3_GT }.

### 2.5.1 Transition Removal

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

 f541_0_rec3_GT: − x4 + x5

### 2.5.2 Trivial Cooperation Program

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

### 2.6 SCC Subproblem 6/6

Here we consider the SCC { f493_0_rec4_LT }.

### 2.6.1 Transition Removal

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

 f493_0_rec4_LT: x1

### 2.6.2 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 % (19 real / 0 unknown / 0 assumptions / 19 total proof steps)