# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: f147_0_get_LE, f1_0_main_Load, f80_0_create_LE, __init, f108_0_main_ArrayAccess
• Transitions: (pre-variables and post-variables)  f1_0_main_Load 1 f80_0_create_LE: x1 = _arg1 ∧ x2 = _arg2 ∧ x3 = _arg3 ∧ x4 = _arg4 ∧ x1 = _arg1P ∧ x2 = _arg2P ∧ x3 = _arg3P ∧ x4 = _arg4P ∧ _arg2 = _arg4P ∧ 0 = _arg3P ∧ _arg2 − 1 = _arg2P ∧ 0 ≤ _arg1P − 1 ∧ 0 ≤ _arg1 − 1 ∧ −1 ≤ _arg2 − 1 ∧ _arg1P ≤ _arg1 f80_0_create_LE 2 f108_0_main_ArrayAccess: x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x1 = _x4 ∧ x2 = _x5 ∧ x3 = _x6 ∧ x4 = _x7 ∧ _x3 = _x6 ∧ 0 ≤ _x4 − 1 ∧ 0 ≤ _x − 1 ∧ _x4 ≤ _x ∧ _x2 ≤ _x5 − 1 ∧ 0 ≤ _x2 − 1 ∧ _x1 ≤ 0 f80_0_create_LE 3 f108_0_main_ArrayAccess: x1 = _x8 ∧ x2 = _x9 ∧ x3 = _x10 ∧ x4 = _x11 ∧ x1 = _x12 ∧ x2 = _x13 ∧ x3 = _x14 ∧ x4 = _x15 ∧ _x11 = _x14 ∧ 1 = _x13 ∧ 0 ≤ _x12 − 1 ∧ 0 ≤ _x8 − 1 ∧ _x9 ≤ 0 ∧ _x12 ≤ _x8 f80_0_create_LE 4 f80_0_create_LE: x1 = _x16 ∧ x2 = _x17 ∧ x3 = _x18 ∧ x4 = _x19 ∧ x1 = _x20 ∧ x2 = _x21 ∧ x3 = _x22 ∧ x4 = _x24 ∧ _x19 = _x24 ∧ _x17 − 1 = _x21 ∧ 0 ≤ _x20 − 1 ∧ 0 ≤ _x16 − 1 ∧ 0 ≤ _x17 − 1 ∧ _x20 ≤ _x16 f108_0_main_ArrayAccess 5 f147_0_get_LE: x1 = _x25 ∧ x2 = _x26 ∧ x3 = _x28 ∧ x4 = _x29 ∧ x1 = _x31 ∧ x2 = _x32 ∧ x3 = _x33 ∧ x4 = _x34 ∧ 0 ≤ _x28 − 1 ∧ −1 ≤ _x35 − 1 ∧ 0 ≤ _x25 − 1 ∧ _x35 − 1 = _x31 ∧ _x26 = _x32 f147_0_get_LE 6 f147_0_get_LE: x1 = _x36 ∧ x2 = _x37 ∧ x3 = _x38 ∧ x4 = _x39 ∧ x1 = _x40 ∧ x2 = _x41 ∧ x3 = _x42 ∧ x4 = _x43 ∧ −1 ≤ _x37 − 1 ∧ 0 ≤ _x44 − 1 ∧ _x44 ≤ _x37 − 1 ∧ 0 ≤ _x36 − 1 ∧ _x44 ≤ _x41 − 1 ∧ _x36 − 1 = _x40 f147_0_get_LE 7 f147_0_get_LE: x1 = _x45 ∧ x2 = _x46 ∧ x3 = _x47 ∧ x4 = _x48 ∧ x1 = _x49 ∧ x2 = _x50 ∧ x3 = _x51 ∧ x4 = _x52 ∧ 0 ≤ _x45 − 1 ∧ _x53 ≤ _x46 − 1 ∧ −1 ≤ _x46 − 1 ∧ _x45 − 1 = _x49 ∧ 1 = _x50 __init 8 f1_0_main_Load: x1 = _x54 ∧ x2 = _x55 ∧ x3 = _x56 ∧ x4 = _x57 ∧ x1 = _x58 ∧ x2 = _x59 ∧ x3 = _x60 ∧ x4 = _x61 ∧ 0 ≤ 0

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 f147_0_get_LE f147_0_get_LE f147_0_get_LE: 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 f80_0_create_LE f80_0_create_LE f80_0_create_LE: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 __init __init __init: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 f108_0_main_ArrayAccess f108_0_main_ArrayAccess f108_0_main_ArrayAccess: 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 { f80_0_create_LE }.

### 2.1.1 Transition Removal

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

 f80_0_create_LE: 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 2/2

Here we consider the SCC { f147_0_get_LE }.

### 2.2.1 Transition Removal

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

 f147_0_get_LE: 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)