# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: f1_0_main_Load', f723_0_init_GE', f723_0_init_GE, f1_0_main_Load, f873_0_init_GE, f1037_0_imprimer_GE, __init, f1074_0_imprimer_GE
• Transitions: (pre-variables and post-variables)  f1_0_main_Load 1 f1_0_main_Load': x1 = _arg1 ∧ x2 = _arg2 ∧ x3 = _arg3 ∧ x4 = _arg4 ∧ x1 = _arg1P ∧ x2 = _arg2P ∧ x3 = _arg3P ∧ x4 = _arg4P ∧ 0 ≤ 2⋅_x39 − 1 ∧ 0 ≤ _arg2 − 1 ∧ 2⋅_x39 + 1 − 2⋅_x40 = 1 ∧ −1 ≤ _x39 − 1 ∧ 0 ≤ _arg1 − 1 ∧ 2 ≤ _x41 − 1 ∧ _arg1 = _arg1P ∧ _arg2 = _arg2P f1_0_main_Load' 2 f723_0_init_GE: x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x1 = _x4 ∧ x2 = _x5 ∧ x3 = _x6 ∧ x4 = _x7 ∧ 0 ≤ 2⋅_x8 − 1 ∧ 0 ≤ _x1 − 1 ∧ 2⋅_x8 + 1 − 2⋅_x9 = 1 ∧ −1 ≤ _x8 − 1 ∧ 0 ≤ _x − 1 ∧ 2 ≤ _x4 − 1 ∧ 2⋅_x8 + 1 − 2⋅_x9 ≤ 1 ∧ 0 ≤ 2⋅_x8 + 1 − 2⋅_x9 ∧ 0 = _x5 ∧ 2⋅_x8 + 1 = _x6 f723_0_init_GE 3 f873_0_init_GE: x1 = _x10 ∧ x2 = _x11 ∧ x3 = _x12 ∧ x4 = _x13 ∧ x1 = _x14 ∧ x2 = _x15 ∧ x3 = _x16 ∧ x4 = _x17 ∧ _x12 = _x17 ∧ 0 = _x16 ∧ _x11 = _x15 ∧ _x12 + 4 ≤ _x10 ∧ 2 ≤ _x14 − 1 ∧ 2 ≤ _x10 − 1 ∧ _x11 ≤ 2 ∧ _x14 ≤ _x10 f873_0_init_GE 4 f723_0_init_GE: x1 = _x19 ∧ x2 = _x20 ∧ x3 = _x21 ∧ x4 = _x22 ∧ x1 = _x23 ∧ x2 = _x24 ∧ x3 = _x25 ∧ x4 = _x26 ∧ _x22 = _x25 ∧ _x20 + 1 = _x24 ∧ _x22 + 4 ≤ _x19 ∧ 2 ≤ _x23 − 1 ∧ 2 ≤ _x19 − 1 ∧ 2 ≤ _x21 − 1 ∧ _x23 ≤ _x19 f873_0_init_GE 5 f873_0_init_GE: x1 = _x27 ∧ x2 = _x28 ∧ x3 = _x29 ∧ x4 = _x30 ∧ x1 = _x31 ∧ x2 = _x32 ∧ x3 = _x33 ∧ x4 = _x34 ∧ _x28 ≤ _x30 − 1 ∧ _x29 ≤ 2 ∧ _x29 ≤ _x35 − 1 ∧ −1 ≤ _x35 − 1 ∧ _x31 ≤ _x27 ∧ 2 ≤ _x27 − 1 ∧ 2 ≤ _x31 − 1 ∧ _x30 + 4 ≤ _x27 ∧ _x28 = _x32 ∧ _x29 + 1 = _x33 ∧ _x30 = _x34 f723_0_init_GE 6 f723_0_init_GE': x1 = _x36 ∧ x2 = _x37 ∧ x3 = _x38 ∧ x4 = _x42 ∧ x1 = _x43 ∧ x2 = _x44 ∧ x3 = _x47 ∧ x4 = _x48 ∧ 1 ≤ _x38 − 1 ∧ 2 ≤ _x37 − 1 ∧ _x49 ≤ _x38 − 1 ∧ _x53 ≤ _x38 − 1 ∧ _x54 ≤ _x36 ∧ 2 ≤ _x36 − 1 ∧ 2 ≤ _x54 − 1 ∧ _x38 + 4 ≤ _x36 ∧ _x36 = _x43 ∧ _x37 = _x44 ∧ _x38 = _x47 f723_0_init_GE' 7 f1037_0_imprimer_GE: x1 = _x55 ∧ x2 = _x56 ∧ x3 = _x59 ∧ x4 = _x60 ∧ x1 = _x61 ∧ x2 = _x62 ∧ x3 = _x63 ∧ x4 = _x64 ∧ 1 ≤ _x59 − 1 ∧ 2 ≤ _x56 − 1 ∧ _x65 ≤ _x59 − 1 ∧ _x66 ≤ _x59 − 1 ∧ _x61 ≤ _x55 ∧ 2 ≤ _x55 − 1 ∧ 2 ≤ _x61 − 1 ∧ _x59 + 4 ≤ _x55 ∧ 0 ≤ _x59 − 2⋅_x65 ∧ _x59 − 2⋅_x65 ≤ 1 ∧ _x59 − 2⋅_x66 ≤ 1 ∧ 0 ≤ _x59 − 2⋅_x66 ∧ 0 = _x62 ∧ _x59 = _x63 f1037_0_imprimer_GE 8 f1074_0_imprimer_GE: x1 = _x67 ∧ x2 = _x68 ∧ x3 = _x69 ∧ x4 = _x70 ∧ x1 = _x71 ∧ x2 = _x72 ∧ x3 = _x73 ∧ x4 = _x74 ∧ _x69 = _x74 ∧ 0 = _x73 ∧ _x68 = _x72 ∧ _x69 + 4 ≤ _x67 ∧ 2 ≤ _x71 − 1 ∧ 2 ≤ _x67 − 1 ∧ _x68 ≤ _x69 − 1 ∧ _x71 ≤ _x67 f1074_0_imprimer_GE 9 f1037_0_imprimer_GE: x1 = _x75 ∧ x2 = _x76 ∧ x3 = _x77 ∧ x4 = _x78 ∧ x1 = _x79 ∧ x2 = _x80 ∧ x3 = _x81 ∧ x4 = _x82 ∧ _x78 = _x81 ∧ _x76 + 1 = _x80 ∧ _x78 + 4 ≤ _x75 ∧ 2 ≤ _x79 − 1 ∧ 2 ≤ _x75 − 1 ∧ _x78 ≤ _x77 ∧ _x79 ≤ _x75 f1074_0_imprimer_GE 10 f1074_0_imprimer_GE: x1 = _x83 ∧ x2 = _x84 ∧ x3 = _x85 ∧ x4 = _x86 ∧ x1 = _x87 ∧ x2 = _x88 ∧ x3 = _x89 ∧ x4 = _x90 ∧ _x86 = _x90 ∧ _x85 + 1 = _x89 ∧ _x84 = _x88 ∧ _x86 + 4 ≤ _x83 ∧ 2 ≤ _x87 − 1 ∧ 2 ≤ _x83 − 1 ∧ _x85 ≤ _x86 − 1 ∧ _x87 ≤ _x83 __init 11 f1_0_main_Load: x1 = _x91 ∧ x2 = _x92 ∧ x3 = _x93 ∧ x4 = _x94 ∧ x1 = _x95 ∧ x2 = _x96 ∧ x3 = _x97 ∧ x4 = _x98 ∧ 0 ≤ 0

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 f1_0_main_Load' f1_0_main_Load' f1_0_main_Load': x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 f723_0_init_GE' f723_0_init_GE' f723_0_init_GE': x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 f723_0_init_GE f723_0_init_GE f723_0_init_GE: 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 f873_0_init_GE f873_0_init_GE f873_0_init_GE: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 f1037_0_imprimer_GE f1037_0_imprimer_GE f1037_0_imprimer_GE: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 __init __init __init: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 f1074_0_imprimer_GE f1074_0_imprimer_GE f1074_0_imprimer_GE: 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 { f723_0_init_GE, f873_0_init_GE }.

### 2.1.1 Transition Removal

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

 f723_0_init_GE: 2 − x2 f873_0_init_GE: 1 − x2

### 2.1.2 Transition Removal

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

 f873_0_init_GE: 0 f723_0_init_GE: −1

### 2.1.3 Transition Removal

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

 f873_0_init_GE: 2 − x3

### 2.1.4 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 { f1037_0_imprimer_GE, f1074_0_imprimer_GE }.

### 2.2.1 Transition Removal

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

 f1037_0_imprimer_GE: 4⋅x3 − 4⋅x2 + 2 f1074_0_imprimer_GE: −4⋅x2 + 4⋅x4 + 1

### 2.2.2 Transition Removal

We remove transition 9 using the following ranking functions, which are bounded by 0.

 f1074_0_imprimer_GE: 0 f1037_0_imprimer_GE: −1

### 2.2.3 Transition Removal

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

 f1074_0_imprimer_GE: x4 − x3

### 2.2.4 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 % (12 real / 0 unknown / 0 assumptions / 12 total proof steps)