# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: f1_0_main_ConstantStackPush, f375_0_dif_GE, f484_0_dif_GE, __init
• Transitions: (pre-variables and post-variables)  f1_0_main_ConstantStackPush 1 f375_0_dif_GE: x1 = _arg1 ∧ x2 = _arg2 ∧ x3 = _arg3 ∧ x4 = _arg4 ∧ x5 = _arg5 ∧ x1 = _arg1P ∧ x2 = _arg2P ∧ x3 = _arg3P ∧ x4 = _arg4P ∧ x5 = _arg5P ∧ 0 = _arg3P ∧ 0 = _arg2P ∧ 0 = _arg1P f375_0_dif_GE 2 f484_0_dif_GE: x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x5 = _x4 ∧ x1 = _x5 ∧ x2 = _x6 ∧ x3 = _x7 ∧ x4 = _x8 ∧ x5 = _x9 ∧ 0 = _x9 ∧ 0 = _x8 ∧ 0 = _x7 ∧ _x1 = _x6 ∧ _x = _x5 ∧ _x1 = _x2 ∧ _x1 ≤ 19 f484_0_dif_GE 3 f375_0_dif_GE: x1 = _x10 ∧ x2 = _x11 ∧ x3 = _x12 ∧ x4 = _x13 ∧ x5 = _x14 ∧ x1 = _x15 ∧ x2 = _x16 ∧ x3 = _x17 ∧ x4 = _x18 ∧ x5 = _x19 ∧ _x11 + 1 = _x17 ∧ _x11 + 1 = _x16 ∧ _x10 = _x15 ∧ _x13 = _x14 ∧ 1 = _x12 ∧ 19 ≤ _x13 − 1 f484_0_dif_GE 4 f375_0_dif_GE: x1 = _x20 ∧ x2 = _x21 ∧ x3 = _x22 ∧ x4 = _x23 ∧ x5 = _x24 ∧ x1 = _x25 ∧ x2 = _x26 ∧ x3 = _x27 ∧ x4 = _x28 ∧ x5 = _x31 ∧ _x21 + 1 = _x27 ∧ _x21 + 1 = _x26 ∧ _x20 = _x25 ∧ _x23 = _x24 ∧ 1 = _x22 ∧ _x23 ≤ 19 f484_0_dif_GE 5 f484_0_dif_GE: x1 = _x32 ∧ x2 = _x33 ∧ x3 = _x34 ∧ x4 = _x35 ∧ x5 = _x36 ∧ x1 = _x37 ∧ x2 = _x38 ∧ x3 = _x39 ∧ x4 = _x40 ∧ x5 = _x41 ∧ _x33 ≤ 19 ∧ _x42 ≤ _x43 − 1 ∧ _x35 ≤ 19 ∧ 0 = _x34 ∧ _x35 = _x36 ∧ _x32 = _x37 ∧ _x33 = _x38 ∧ 0 = _x39 ∧ _x35 + 1 = _x40 ∧ _x35 + 1 = _x41 f484_0_dif_GE 6 f375_0_dif_GE: x1 = _x44 ∧ x2 = _x45 ∧ x3 = _x46 ∧ x4 = _x47 ∧ x5 = _x48 ∧ x1 = _x49 ∧ x2 = _x50 ∧ x3 = _x51 ∧ x4 = _x52 ∧ x5 = _x53 ∧ _x45 + 1 = _x51 ∧ _x45 + 1 = _x50 ∧ _x44 + 1 = _x49 ∧ _x47 = _x48 ∧ 0 = _x46 ∧ 19 ≤ _x47 − 1 ∧ _x45 ≤ 19 ∧ _x44 ≤ 19 f484_0_dif_GE 7 f484_0_dif_GE: x1 = _x54 ∧ x2 = _x55 ∧ x3 = _x56 ∧ x4 = _x57 ∧ x5 = _x58 ∧ x1 = _x59 ∧ x2 = _x60 ∧ x3 = _x61 ∧ x4 = _x62 ∧ x5 = _x63 ∧ _x57 = _x63 ∧ _x57 = _x62 ∧ 1 = _x61 ∧ _x55 = _x60 ∧ _x54 = _x59 ∧ _x57 = _x58 ∧ 0 = _x56 ∧ _x55 ≤ 19 ∧ _x57 ≤ 19 __init 8 f1_0_main_ConstantStackPush: x1 = _x64 ∧ x2 = _x65 ∧ x3 = _x66 ∧ x4 = _x67 ∧ x5 = _x68 ∧ x1 = _x69 ∧ x2 = _x70 ∧ x3 = _x71 ∧ x4 = _x72 ∧ x5 = _x73 ∧ 0 ≤ 0

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 f1_0_main_ConstantStackPush f1_0_main_ConstantStackPush f1_0_main_ConstantStackPush: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 f375_0_dif_GE f375_0_dif_GE f375_0_dif_GE: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 f484_0_dif_GE f484_0_dif_GE f484_0_dif_GE: 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 1 SCC(s) of the program graph.

### 2.1 SCC Subproblem 1/1

Here we consider the SCC { f375_0_dif_GE, f484_0_dif_GE }.

### 2.1.1 Transition Removal

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

 f375_0_dif_GE: 19 − x2 f484_0_dif_GE: 19 − x2 − x3

### 2.1.2 Transition Removal

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

 f375_0_dif_GE: 19 − x2 f484_0_dif_GE: 18 − x2

### 2.1.3 Transition Removal

We remove transitions 4, 3 using the following ranking functions, which are bounded by 0.

 f484_0_dif_GE: −1 + x3 f375_0_dif_GE: −1

### 2.1.4 Transition Removal

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

 f484_0_dif_GE: 19 + 0⋅x3 − x5

### 2.1.5 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)