# LTS Termination Proof

by T2Cert

## Input

Integer Transition System
• Initial Location: 2
• Transitions: (pre-variables and post-variables)  0 0 1: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − arg2P ≤ 0 ∧ − arg2 ≤ 0 ∧ − arg1P ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 1 1 1: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − arg2 ≤ 0 ∧ −1 − arg1P + arg2 ≤ 0 ∧ 1 + arg1P − arg2 ≤ 0 ∧ arg1 − arg2P ≤ 0 ∧ − arg1 + arg2P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 2 2 0: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 1 3 1: − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0
and for every transition t, a duplicate t is considered.

### 2 Transition Removal

We remove transitions 0, 2 using the following ranking functions, which are bounded by −11.

 2: 0 0: 0 1: 0 2: −4 0: −5 1: −6 1_var_snapshot: −6 1*: −6
Hints:
 4 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0] ] 1 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 0 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 2 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0] ]

### 3 Location Addition

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

1* 6 1: arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

### 4 Location Addition

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

1 4 1_var_snapshot: arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

### 5 SCC Decomposition

We consider subproblems for each of the 1 SCC(s) of the program graph.

### 5.1 SCC Subproblem 1/1

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

### 5.1.1 Splitting Cut-Point Transitions

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

### 5.1.1.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 3.

### 5.1.1.1.1 Fresh Variable Addition

The new variable __snapshot_1_arg2P is introduced. The transition formulas are extended as follows:

 4: __snapshot_1_arg2P ≤ arg2P ∧ arg2P ≤ __snapshot_1_arg2P 6: __snapshot_1_arg2P ≤ __snapshot_1_arg2P ∧ __snapshot_1_arg2P ≤ __snapshot_1_arg2P 1: __snapshot_1_arg2P ≤ __snapshot_1_arg2P ∧ __snapshot_1_arg2P ≤ __snapshot_1_arg2P

### 5.1.1.1.2 Fresh Variable Addition

The new variable __snapshot_1_arg2 is introduced. The transition formulas are extended as follows:

 4: __snapshot_1_arg2 ≤ arg2 ∧ arg2 ≤ __snapshot_1_arg2 6: __snapshot_1_arg2 ≤ __snapshot_1_arg2 ∧ __snapshot_1_arg2 ≤ __snapshot_1_arg2 1: __snapshot_1_arg2 ≤ __snapshot_1_arg2 ∧ __snapshot_1_arg2 ≤ __snapshot_1_arg2

### 5.1.1.1.3 Fresh Variable Addition

The new variable __snapshot_1_arg1P is introduced. The transition formulas are extended as follows:

 4: __snapshot_1_arg1P ≤ arg1P ∧ arg1P ≤ __snapshot_1_arg1P 6: __snapshot_1_arg1P ≤ __snapshot_1_arg1P ∧ __snapshot_1_arg1P ≤ __snapshot_1_arg1P 1: __snapshot_1_arg1P ≤ __snapshot_1_arg1P ∧ __snapshot_1_arg1P ≤ __snapshot_1_arg1P

### 5.1.1.1.4 Fresh Variable Addition

The new variable __snapshot_1_arg1 is introduced. The transition formulas are extended as follows:

 4: __snapshot_1_arg1 ≤ arg1 ∧ arg1 ≤ __snapshot_1_arg1 6: __snapshot_1_arg1 ≤ __snapshot_1_arg1 ∧ __snapshot_1_arg1 ≤ __snapshot_1_arg1 1: __snapshot_1_arg1 ≤ __snapshot_1_arg1 ∧ __snapshot_1_arg1 ≤ __snapshot_1_arg1

### 5.1.1.1.5 Invariant Updates

The following invariants are asserted.

 0: TRUE 1: arg1 − arg1P + arg2 − arg2P ≤ 0 ∧ − arg1P − arg2P ≤ 0 ∧ − arg2 ≤ 0 ∨ arg1 − arg1P + arg2 − arg2P ≤ 0 ∧ −1 − arg1P + arg2 − arg2P ≤ 0 2: TRUE 1: arg1 − arg1P + arg2 − arg2P ≤ 0 ∧ − arg1P − arg2P ≤ 0 ∧ − arg2 ≤ 0 ∨ arg1 − arg1P + arg2 − arg2P ≤ 0 ∧ −1 − arg1P + arg2 − arg2P ≤ 0 ∧ 1 − __snapshot_1_arg1P − __snapshot_1_arg2P + arg1P + arg2P ≤ 0 ∧ − __snapshot_1_arg1P − __snapshot_1_arg2P ≤ 0 ∨ arg1 − arg1P + arg2 − arg2P ≤ 0 ∧ −1 − arg1P + arg2 − arg2P ≤ 0 1_var_snapshot: − __snapshot_1_arg1P − __snapshot_1_arg2P + arg1 + arg2 ≤ 0 ∧ − __snapshot_1_arg1P − __snapshot_1_arg2P ≤ 0 ∧ − arg2 ≤ 0 ∨ − __snapshot_1_arg1P − __snapshot_1_arg2P + arg1 + arg2 ≤ 0 ∧ −1 − __snapshot_1_arg1P − __snapshot_1_arg2P + arg2 ≤ 0 1*: arg1 − arg1P + arg2 − arg2P ≤ 0 ∧ −1 − arg1P + arg2 − arg2P ≤ 0 ∧ 1 − __snapshot_1_arg1P − __snapshot_1_arg2P + arg1P + arg2P ≤ 0 ∧ − __snapshot_1_arg1P − __snapshot_1_arg2P ≤ 0

The invariants are proved as follows.

### IMPACT Invariant Proof

• nodes (location) invariant:  0 (2) TRUE 1 (0) TRUE 2 (1) arg1 − arg1P + arg2 − arg2P ≤ 0 ∧ − arg1P − arg2P ≤ 0 ∧ − arg2 ≤ 0 3 (1) arg1 − arg1P + arg2 − arg2P ≤ 0 ∧ −1 − arg1P + arg2 − arg2P ≤ 0 4 (1) arg1 − arg1P + arg2 − arg2P ≤ 0 ∧ − arg1P − arg2P ≤ 0 ∧ − arg2 ≤ 0 5 (1_var_snapshot) − __snapshot_1_arg1P − __snapshot_1_arg2P + arg1 + arg2 ≤ 0 ∧ − __snapshot_1_arg1P − __snapshot_1_arg2P ≤ 0 ∧ − arg2 ≤ 0 12 (1*) arg1 − arg1P + arg2 − arg2P ≤ 0 ∧ −1 − arg1P + arg2 − arg2P ≤ 0 ∧ 1 − __snapshot_1_arg1P − __snapshot_1_arg2P + arg1P + arg2P ≤ 0 ∧ − __snapshot_1_arg1P − __snapshot_1_arg2P ≤ 0 13 (1) arg1 − arg1P + arg2 − arg2P ≤ 0 ∧ −1 − arg1P + arg2 − arg2P ≤ 0 ∧ 1 − __snapshot_1_arg1P − __snapshot_1_arg2P + arg1P + arg2P ≤ 0 ∧ − __snapshot_1_arg1P − __snapshot_1_arg2P ≤ 0 14 (1_var_snapshot) − __snapshot_1_arg1P − __snapshot_1_arg2P + arg1 + arg2 ≤ 0 ∧ −1 − __snapshot_1_arg1P − __snapshot_1_arg2P + arg2 ≤ 0 15 (1_var_snapshot) − __snapshot_1_arg1P − __snapshot_1_arg2P + arg1 + arg2 ≤ 0 ∧ −1 − __snapshot_1_arg1P − __snapshot_1_arg2P + arg2 ≤ 0 23 (1) arg1 − arg1P + arg2 − arg2P ≤ 0 ∧ −1 − arg1P + arg2 − arg2P ≤ 0 24 (1) arg1 − arg1P + arg2 − arg2P ≤ 0 ∧ −1 − arg1P + arg2 − arg2P ≤ 0 28 (1*) arg1 − arg1P + arg2 − arg2P ≤ 0 ∧ −1 − arg1P + arg2 − arg2P ≤ 0 ∧ 1 − __snapshot_1_arg1P − __snapshot_1_arg2P + arg1P + arg2P ≤ 0 ∧ − __snapshot_1_arg1P − __snapshot_1_arg2P ≤ 0 37 (1_var_snapshot) − __snapshot_1_arg1P − __snapshot_1_arg2P + arg1 + arg2 ≤ 0 ∧ −1 − __snapshot_1_arg1P − __snapshot_1_arg2P + arg2 ≤ 0
• initial node: 0
• cover edges:
14 → 15 Hint: distribute conclusion [1, 0] [0, 1]
23 → 3 Hint: distribute conclusion [1, 0] [0, 1]
28 → 12 Hint: distribute conclusion [1, 0, 0, 0] [0, 1, 0, 0] [0, 0, 1, 0] [0, 0, 0, 1]
37 → 15 Hint: distribute conclusion [1, 0] [0, 1]
• transition edges:
0 2 1 Hint: auto
1 0 2 Hint: distribute conclusion [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0] [0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0] [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1]
2 1 3 Hint: distribute conclusion [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0] [0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0]
2 3 4 Hint: distribute conclusion [1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0] [0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0] [0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0]
3 1 23 Hint: distribute conclusion [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0] [0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0]
3 3 24 Hint: distribute conclusion [1, 0, 0, 1, 1, 0, 0, 1, 1, 0] [0, 1, 0, 1, 1, 0, 0, 1, 0, 0]
4 4 5 Hint: distribute conclusion [1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0] [0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0] [0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
5 1 12 Hint: distribute conclusion [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0] [0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0] [1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0] [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0]
12 6 13 Hint: distribute conclusion [1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0] [0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] [0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0] [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0]
13 4 14 Hint: distribute conclusion [1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0] [0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0]
13 4 15 Hint: distribute conclusion [1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0] [0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0]
15 1 28 Hint: distribute conclusion [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0] [0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0] [1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0] [0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0]
24 4 37 Hint: distribute conclusion [1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0] [0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0]

### 5.1.1.1.6 Transition Removal

We remove transition 6 using the following ranking functions, which are bounded by −2.

 1: arg1P + arg2P 1_var_snapshot: __snapshot_1_arg1P + __snapshot_1_arg2P 1*: __snapshot_1_arg1P + __snapshot_1_arg2P
Hints:
4 distribute assertion
 lexWeak[ [0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0] ] lexWeak[ [0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0] ] lexWeak[ [0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0] ]
6 lexStrict[ [0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
1 distribute assertion
 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0] ] lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0] ]

### 5.1.1.1.7 Transition Removal

We remove transition 4 using the following ranking functions, which are bounded by −5.

 1: −1 1_var_snapshot: −2 1*: −3
Hints:
4 distribute assertion
 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
1 distribute assertion
 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

### 5.1.1.1.8 Splitting Cut-Point Transitions

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

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