# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: f1_0_main_ConstantStackPush, f543_0_sort_GE, f458_0_sort_GE, __init
• Transitions: (pre-variables and post-variables)  f1_0_main_ConstantStackPush 1 f458_0_sort_GE: x1 = _arg1 ∧ x2 = _arg2 ∧ x3 = _arg3 ∧ x1 = _arg1P ∧ x2 = _arg2P ∧ x3 = _arg3P ∧ 1 = _arg1P f458_0_sort_GE 2 f543_0_sort_GE: x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x1 = _x3 ∧ x2 = _x4 ∧ x3 = _x5 ∧ 100 − _x = _x5 ∧ 0 = _x4 ∧ _x = _x3 ∧ _x ≤ 99 ∧ 0 ≤ _x − 1 f543_0_sort_GE 3 f458_0_sort_GE: x1 = _x6 ∧ x2 = _x9 ∧ x3 = _x10 ∧ x1 = _x11 ∧ x2 = _x14 ∧ x3 = _x15 ∧ _x6 + 1 = _x11 ∧ _x10 ≤ _x9 f543_0_sort_GE 4 f543_0_sort_GE: x1 = _x16 ∧ x2 = _x17 ∧ x3 = _x18 ∧ x1 = _x19 ∧ x2 = _x20 ∧ x3 = _x21 ∧ _x17 ≤ 99 ∧ _x17 ≤ _x18 − 1 ∧ −1 ≤ _x17 − 1 ∧ _x17 ≤ 98 ∧ _x22 ≤ _x23 ∧ 0 ≤ _x16 − 1 ∧ _x16 = _x19 ∧ _x17 + 1 = _x20 ∧ 100 − _x16 = _x21 f543_0_sort_GE 5 f543_0_sort_GE: x1 = _x24 ∧ x2 = _x25 ∧ x3 = _x26 ∧ x1 = _x27 ∧ x2 = _x28 ∧ x3 = _x29 ∧ _x25 ≤ 99 ∧ _x25 ≤ _x26 − 1 ∧ −1 ≤ _x25 − 1 ∧ _x25 ≤ 98 ∧ 0 ≤ _x24 − 1 ∧ _x30 ≤ _x31 − 1 ∧ _x24 = _x27 ∧ _x25 + 1 = _x28 ∧ 100 − _x24 = _x29 __init 6 f1_0_main_ConstantStackPush: x1 = _x32 ∧ x2 = _x33 ∧ x3 = _x34 ∧ x1 = _x35 ∧ x2 = _x36 ∧ x3 = _x37 ∧ 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 f543_0_sort_GE f543_0_sort_GE f543_0_sort_GE: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 f458_0_sort_GE f458_0_sort_GE f458_0_sort_GE: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 __init __init __init: 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 1 SCC(s) of the program graph.

### 2.1 SCC Subproblem 1/1

Here we consider the SCC { f543_0_sort_GE, f458_0_sort_GE }.

### 2.1.1 Transition Removal

We remove transition 2 using the following ranking functions, which are bounded by −196.

 f458_0_sort_GE: −2⋅x1 + 2 f543_0_sort_GE: −2⋅x1 + 1

### 2.1.2 Transition Removal

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

 f543_0_sort_GE: 0 f458_0_sort_GE: −1

### 2.1.3 Transition Removal

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

 f543_0_sort_GE: 98 − x2

### 2.1.4 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)