# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: f411_0_slide95_EQ', f411_0_slide95_EQ, f1_0_main_Load, f196_0_create_LE, __init, f234_0_slide95_FieldAccess
• Transitions: (pre-variables and post-variables)  f1_0_main_Load 1 f234_0_slide95_FieldAccess: x1 = _arg1 ∧ x2 = _arg2 ∧ x3 = _arg3 ∧ x1 = _arg1P ∧ x2 = _arg2P ∧ x3 = _arg3P ∧ −1 ≤ _x4 − 1 ∧ 1 ≤ _arg2 − 1 ∧ −1 ≤ _arg1P − 1 ∧ _arg2P ≤ _x5 − 1 ∧ −1 ≤ _x5 − 1 ∧ 0 ≤ _arg1 − 1 f1_0_main_Load 2 f234_0_slide95_FieldAccess: x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x1 = _x3 ∧ x2 = _x6 ∧ x3 = _x7 ∧ −1 ≤ _x8 − 1 ∧ 1 ≤ _x1 − 1 ∧ _x6 ≤ 0 ∧ −1 ≤ _x3 − 1 ∧ 0 ≤ _x − 1 f234_0_slide95_FieldAccess 3 f411_0_slide95_EQ: x1 = _x9 ∧ x2 = _x11 ∧ x3 = _x12 ∧ x1 = _x13 ∧ x2 = _x14 ∧ x3 = _x15 ∧ _x11 = _x15 ∧ _x11 = _x14 ∧ _x9 = _x13 ∧ 0 ≤ _x11 − 1 f411_0_slide95_EQ 4 f411_0_slide95_EQ': x1 = _x16 ∧ x2 = _x17 ∧ x3 = _x18 ∧ x1 = _x19 ∧ x2 = _x20 ∧ x3 = _x21 ∧ −1 ≤ _x22 − 1 ∧ 0 ≤ _x17 − 1 ∧ _x22 ≤ _x23 − 1 ∧ −1 ≤ _x18 − 1 ∧ _x22 ≤ _x18 − 1 ∧ _x22 ≤ _x24 − 1 ∧ _x25 ≤ _x17 − 1 ∧ _x22 ≤ _x25 − 1 ∧ _x22 ≤ _x26 − 1 ∧ _x27 ≤ _x16 ∧ _x16 − 2⋅_x28 = 0 ∧ _x16 = _x19 ∧ _x17 = _x20 ∧ _x18 = _x21 f411_0_slide95_EQ' 5 f411_0_slide95_EQ: x1 = _x29 ∧ x2 = _x30 ∧ x3 = _x31 ∧ x1 = _x32 ∧ x2 = _x33 ∧ x3 = _x34 ∧ −1 ≤ _x35 − 1 ∧ 0 ≤ _x30 − 1 ∧ _x35 ≤ _x34 − 1 ∧ −1 ≤ _x31 − 1 ∧ _x35 ≤ _x31 − 1 ∧ _x35 ≤ _x38 − 1 ∧ _x33 ≤ _x30 − 1 ∧ _x35 ≤ _x33 − 1 ∧ _x35 ≤ _x46 − 1 ∧ _x29 − 2⋅_x47 = 0 ∧ _x32 ≤ _x29 ∧ 0 ≤ _x29 − 2⋅_x47 ∧ _x29 − 2⋅_x47 ≤ 1 ∧ _x29 − 2⋅_x32 ≤ 1 ∧ 0 ≤ _x29 − 2⋅_x32 f411_0_slide95_EQ 6 f411_0_slide95_EQ': x1 = _x48 ∧ x2 = _x49 ∧ x3 = _x50 ∧ x1 = _x51 ∧ x2 = _x56 ∧ x3 = _x57 ∧ −1 ≤ _x50 − 1 ∧ 0 ≤ _x49 − 1 ∧ _x58 ≤ _x50 − 1 ∧ _x64 ≤ _x49 − 1 ∧ _x48 − 2⋅_x65 = 1 ∧ _x66 ≤ _x48 ∧ −1 ≤ _x64 − 1 ∧ _x67 ≤ _x64 ∧ 0 ≤ _x58 − 1 ∧ _x48 = _x51 ∧ _x49 = _x56 ∧ _x50 = _x57 f411_0_slide95_EQ' 7 f411_0_slide95_EQ: x1 = _x68 ∧ x2 = _x69 ∧ x3 = _x72 ∧ x1 = _x73 ∧ x2 = _x74 ∧ x3 = _x78 ∧ −1 ≤ _x72 − 1 ∧ 0 ≤ _x69 − 1 ∧ _x78 ≤ _x72 − 1 ∧ _x79 ≤ _x69 − 1 ∧ _x68 − 2⋅_x80 = 1 ∧ _x73 ≤ _x68 ∧ −1 ≤ _x79 − 1 ∧ 0 ≤ _x78 − 1 ∧ _x74 ≤ _x79 ∧ 0 ≤ _x68 − 2⋅_x80 ∧ _x68 − 2⋅_x80 ≤ 1 ∧ _x68 − 2⋅_x73 ≤ 1 ∧ 0 ≤ _x68 − 2⋅_x73 f411_0_slide95_EQ 8 f411_0_slide95_EQ': x1 = _x81 ∧ x2 = _x82 ∧ x3 = _x84 ∧ x1 = _x85 ∧ x2 = _x86 ∧ x3 = _x87 ∧ −1 ≤ _x84 − 1 ∧ 0 ≤ _x82 − 1 ∧ _x88 ≤ _x84 − 1 ∧ _x88 ≤ _x82 − 1 ∧ _x81 − 2⋅_x89 = 1 ∧ 0 ≤ _x88 − 1 ∧ _x90 ≤ _x81 ∧ _x81 = _x85 ∧ _x82 = _x86 ∧ _x84 = _x87 f411_0_slide95_EQ' 9 f411_0_slide95_EQ: x1 = _x91 ∧ x2 = _x92 ∧ x3 = _x93 ∧ x1 = _x94 ∧ x2 = _x95 ∧ x3 = _x96 ∧ −1 ≤ _x93 − 1 ∧ 0 ≤ _x92 − 1 ∧ _x95 ≤ _x93 − 1 ∧ _x95 ≤ _x92 − 1 ∧ _x91 − 2⋅_x97 = 1 ∧ _x94 ≤ _x91 ∧ 0 ≤ _x95 − 1 ∧ 0 ≤ _x91 − 2⋅_x97 ∧ _x91 − 2⋅_x97 ≤ 1 ∧ _x91 − 2⋅_x94 ≤ 1 ∧ 0 ≤ _x91 − 2⋅_x94 ∧ _x95 = _x96 f1_0_main_Load 10 f196_0_create_LE: x1 = _x98 ∧ x2 = _x99 ∧ x3 = _x100 ∧ x1 = _x101 ∧ x2 = _x102 ∧ x3 = _x103 ∧ −1 ≤ _x104 − 1 ∧ 1 ≤ _x99 − 1 ∧ −1 ≤ _x105 − 1 ∧ 0 ≤ _x98 − 1 ∧ _x105 − 1 = _x101 f196_0_create_LE 11 f196_0_create_LE: x1 = _x106 ∧ x2 = _x107 ∧ x3 = _x108 ∧ x1 = _x109 ∧ x2 = _x110 ∧ x3 = _x111 ∧ _x106 − 1 = _x109 ∧ 0 ≤ _x106 − 1 __init 12 f1_0_main_Load: x1 = _x112 ∧ x2 = _x113 ∧ x3 = _x114 ∧ x1 = _x115 ∧ x2 = _x116 ∧ x3 = _x117 ∧ 0 ≤ 0

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 f411_0_slide95_EQ' f411_0_slide95_EQ' f411_0_slide95_EQ': x1 = x1 ∧ x2 = x2 ∧ x3 = x3 f411_0_slide95_EQ f411_0_slide95_EQ f411_0_slide95_EQ: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 f1_0_main_Load f1_0_main_Load f1_0_main_Load: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 f196_0_create_LE f196_0_create_LE f196_0_create_LE: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 __init __init __init: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 f234_0_slide95_FieldAccess f234_0_slide95_FieldAccess f234_0_slide95_FieldAccess: x1 = x1 ∧ x2 = x2 ∧ x3 = x3
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 { f196_0_create_LE }.

### 2.1.1 Transition Removal

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

 f196_0_create_LE: 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 { f411_0_slide95_EQ', f411_0_slide95_EQ }.

### 2.2.1 Transition Removal

We remove transitions 4, 5, 6, 7, 8, 9 using the following ranking functions, which are bounded by 0.

 f411_0_slide95_EQ: 2⋅x2 f411_0_slide95_EQ': 2⋅x2 − 1

### 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 % (7 real / 0 unknown / 0 assumptions / 7 total proof steps)