# LTS Termination Proof

by T2Cert

## Input

Integer Transition System
• Initial Location: 7

## Proof

The following invariants are asserted.

The invariants are proved as follows.

### IMPACT Invariant Proof

• initial node: 7
• cover edges:
• transition edges:  0 0 1 0 1 2 1 4 4 1 5 5 2 2 0 3 3 0 4 6 6 4 7 1 7 8 3

### 2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
and for every transition t, a duplicate t is considered.

### 3 Transition Removal

We remove transitions 0, 3, 5, 6, 8 using the following ranking functions, which are bounded by −19.

 7: 0 3: 0 0: 0 2: 0 1: 0 4: 0 6: 0 5: 0 7: −7 3: −8 0: −9 2: −9 0_var_snapshot: −9 0*: −9 1: −12 4: −12 1_var_snapshot: −12 1*: −12 6: −13 5: −17

The following skip-transition is inserted and corresponding redirections w.r.t. the old location are performed.

The following skip-transition is inserted and corresponding redirections w.r.t. the old location are performed.

The following skip-transition is inserted and corresponding redirections w.r.t. the old location are performed.

The following skip-transition is inserted and corresponding redirections w.r.t. the old location are performed.

### 8 SCC Decomposition

We consider subproblems for each of the 2 SCC(s) of the program graph.

### 8.1 SCC Subproblem 1/2

Here we consider the SCC { 1, 4, 1_var_snapshot, 1* }.

### 8.1.1 Transition Removal

We remove transitions 4, 7 using the following ranking functions, which are bounded by 14.

 1: 4⋅a_243_0 + length_7_post 4: −19 + 4⋅a_243_0 + 2⋅length_7_post 1_var_snapshot: −1 + 4⋅a_243_0 + length_7_post 1*: −16 + 4⋅a_243_0 + 2⋅length_7_post

### 8.1.2 Transition Removal

We remove transitions 17, 19 using the following ranking functions, which are bounded by −1.

 1: 0 4: 0 1_var_snapshot: − length_7_post 1*: len_48_0

### 8.1.3 Splitting Cut-Point Transitions

We consider 1 subproblems corresponding to sets of cut-point transitions as follows.

### 8.1.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 16.

### 8.1.3.1.1 Splitting Cut-Point Transitions

There remain no cut-point transition to consider. Hence the cooperation termination is trivial.

### 8.2 SCC Subproblem 2/2

Here we consider the SCC { 0, 2, 0_var_snapshot, 0* }.

### 8.2.1 Transition Removal

We remove transition 1 using the following ranking functions, which are bounded by −884.

 0: −34 − 52⋅i_8_0 2: −52⋅i_8_0 0_var_snapshot: −52⋅i_8_0 − 3⋅length_7_post 0*: −17 − 52⋅i_8_0

### 8.2.2 Transition Removal

We remove transitions 10, 2 using the following ranking functions, which are bounded by −1.

 0: 0 2: 1 + length_7_0 0_var_snapshot: − length_7_post 0*: length_7_0

### 8.2.3 Transition Removal

We remove transition 12 using the following ranking functions, which are bounded by 16.

 0: 0 2: 0 0_var_snapshot: 0 0*: length_7_post

### 8.2.4 Splitting Cut-Point Transitions

We consider 1 subproblems corresponding to sets of cut-point transitions as follows.

### 8.2.4.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 9.

### 8.2.4.1.1 Splitting Cut-Point Transitions

There remain no cut-point transition to consider. Hence the cooperation termination is trivial.

T2Cert

• version: 1.0