# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: l4, l3, l0, l2
• Transitions: (pre-variables and post-variables)  l0 1 l1: x1 = _Fnew5HAT0 ∧ x2 = _Fold6HAT0 ∧ x3 = ___const_30HAT0 ∧ x4 = _aHAT0 ∧ x5 = _ans8HAT0 ∧ x6 = _i4HAT0 ∧ x7 = _n3HAT0 ∧ x8 = _ret_fib9HAT0 ∧ x9 = _temp7HAT0 ∧ x10 = _tmpHAT0 ∧ x1 = _Fnew5HATpost ∧ x2 = _Fold6HATpost ∧ x3 = ___const_30HATpost ∧ x4 = _aHATpost ∧ x5 = _ans8HATpost ∧ x6 = _i4HATpost ∧ x7 = _n3HATpost ∧ x8 = _ret_fib9HATpost ∧ x9 = _temp7HATpost ∧ x10 = _tmpHATpost ∧ _temp7HAT0 = _temp7HATpost ∧ _n3HAT0 = _n3HATpost ∧ _i4HAT0 = _i4HATpost ∧ _aHAT0 = _aHATpost ∧ ___const_30HAT0 = ___const_30HATpost ∧ _Fold6HAT0 = _Fold6HATpost ∧ _Fnew5HAT0 = _Fnew5HATpost ∧ _tmpHATpost = _ret_fib9HATpost ∧ _ret_fib9HATpost = _ans8HATpost ∧ _ans8HATpost = _Fnew5HAT0 ∧ 1 + _n3HAT0 ≤ _i4HAT0 l0 2 l2: x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x5 = _x4 ∧ x6 = _x5 ∧ x7 = _x6 ∧ x8 = _x7 ∧ x9 = _x8 ∧ x10 = _x9 ∧ x1 = _x10 ∧ x2 = _x11 ∧ x3 = _x12 ∧ x4 = _x13 ∧ x5 = _x14 ∧ x6 = _x15 ∧ x7 = _x16 ∧ x8 = _x17 ∧ x9 = _x18 ∧ x10 = _x19 ∧ _x9 = _x19 ∧ _x7 = _x17 ∧ _x6 = _x16 ∧ _x4 = _x14 ∧ _x3 = _x13 ∧ _x2 = _x12 ∧ _x15 = 1 + _x5 ∧ _x11 = _x18 ∧ _x10 = _x + _x1 ∧ _x18 = _x ∧ _x5 ≤ _x6 l2 3 l0: x1 = _x20 ∧ x2 = _x21 ∧ x3 = _x22 ∧ x4 = _x23 ∧ x5 = _x24 ∧ x6 = _x25 ∧ x7 = _x26 ∧ x8 = _x27 ∧ x9 = _x28 ∧ x10 = _x29 ∧ x1 = _x30 ∧ x2 = _x31 ∧ x3 = _x32 ∧ x4 = _x33 ∧ x5 = _x34 ∧ x6 = _x35 ∧ x7 = _x36 ∧ x8 = _x37 ∧ x9 = _x38 ∧ x10 = _x39 ∧ _x29 = _x39 ∧ _x28 = _x38 ∧ _x27 = _x37 ∧ _x26 = _x36 ∧ _x25 = _x35 ∧ _x24 = _x34 ∧ _x23 = _x33 ∧ _x22 = _x32 ∧ _x21 = _x31 ∧ _x20 = _x30 l3 4 l2: x1 = _x40 ∧ x2 = _x41 ∧ x3 = _x42 ∧ x4 = _x43 ∧ x5 = _x44 ∧ x6 = _x45 ∧ x7 = _x46 ∧ x8 = _x47 ∧ x9 = _x48 ∧ x10 = _x49 ∧ x1 = _x50 ∧ x2 = _x51 ∧ x3 = _x52 ∧ x4 = _x53 ∧ x5 = _x54 ∧ x6 = _x55 ∧ x7 = _x56 ∧ x8 = _x57 ∧ x9 = _x58 ∧ x10 = _x59 ∧ _x49 = _x59 ∧ _x48 = _x58 ∧ _x47 = _x57 ∧ _x44 = _x54 ∧ _x42 = _x52 ∧ _x55 = 2 ∧ _x51 = 0 ∧ _x50 = 1 ∧ _x56 = _x53 ∧ _x53 = _x42 l4 5 l3: x1 = _x60 ∧ x2 = _x61 ∧ x3 = _x62 ∧ x4 = _x63 ∧ x5 = _x64 ∧ x6 = _x65 ∧ x7 = _x66 ∧ x8 = _x67 ∧ x9 = _x68 ∧ x10 = _x69 ∧ x1 = _x70 ∧ x2 = _x71 ∧ x3 = _x72 ∧ x4 = _x73 ∧ x5 = _x74 ∧ x6 = _x75 ∧ x7 = _x76 ∧ x8 = _x77 ∧ x9 = _x78 ∧ x10 = _x79 ∧ _x69 = _x79 ∧ _x68 = _x78 ∧ _x67 = _x77 ∧ _x66 = _x76 ∧ _x65 = _x75 ∧ _x64 = _x74 ∧ _x63 = _x73 ∧ _x62 = _x72 ∧ _x61 = _x71 ∧ _x60 = _x70

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 l4 l4 l4: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 l3 l3 l3: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 l0 l0 l0: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 l2 l2 l2: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10
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 { l0, l2 }.

### 2.1.1 Transition Removal

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

 l0: −2⋅x6 + 2⋅x7 l2: −2⋅x6 + 2⋅x7 + 1

### 2.1.2 Transition Removal

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

 l2: 0 l0: −1

### 2.1.3 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 % (6 real / 0 unknown / 0 assumptions / 6 total proof steps)