# LTS Termination Proof

by T2Cert

## Input

Integer Transition System
• Initial Location: 5
• Transitions: (pre-variables and post-variables)  0 0 1: 36 − counter_0 ≤ 0 ∧ − z_post + z_post ≤ 0 ∧ z_post − z_post ≤ 0 ∧ − z_0 + z_0 ≤ 0 ∧ z_0 − z_0 ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − counter_post + counter_post ≤ 0 ∧ counter_post − counter_post ≤ 0 ∧ − counter_0 + counter_0 ≤ 0 ∧ counter_0 − counter_0 ≤ 0 0 1 2: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ −35 + counter_0 ≤ 0 ∧ −1 − z_0 + z_post ≤ 0 ∧ 1 + z_0 − z_post ≤ 0 ∧ −1 − counter_0 + counter_post ≤ 0 ∧ 1 + counter_0 − counter_post ≤ 0 ∧ counter_0 − counter_post ≤ 0 ∧ − counter_0 + counter_post ≤ 0 ∧ z_0 − z_post ≤ 0 ∧ − z_0 + z_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 3 2 2: 0 ≤ 0 ∧ 0 ≤ 0 ∧ −127 + y_0 ≤ 0 ∧ z_0 − z_post ≤ 0 ∧ − z_0 + z_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − counter_post + counter_post ≤ 0 ∧ counter_post − counter_post ≤ 0 ∧ − counter_0 + counter_0 ≤ 0 ∧ counter_0 − counter_0 ≤ 0 3 3 1: 128 − y_0 ≤ 0 ∧ − z_post + z_post ≤ 0 ∧ z_post − z_post ≤ 0 ∧ − z_0 + z_0 ≤ 0 ∧ z_0 − z_0 ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − counter_post + counter_post ≤ 0 ∧ counter_post − counter_post ≤ 0 ∧ − counter_0 + counter_0 ≤ 0 ∧ counter_0 − counter_0 ≤ 0 2 4 0: − z_post + z_post ≤ 0 ∧ z_post − z_post ≤ 0 ∧ − z_0 + z_0 ≤ 0 ∧ z_0 − z_0 ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − counter_post + counter_post ≤ 0 ∧ counter_post − counter_post ≤ 0 ∧ − counter_0 + counter_0 ≤ 0 ∧ counter_0 − counter_0 ≤ 0 4 5 3: 0 ≤ 0 ∧ 0 ≤ 0 ∧ counter_post ≤ 0 ∧ − counter_post ≤ 0 ∧ counter_0 − counter_post ≤ 0 ∧ − counter_0 + counter_post ≤ 0 ∧ − z_post + z_post ≤ 0 ∧ z_post − z_post ≤ 0 ∧ − z_0 + z_0 ≤ 0 ∧ z_0 − z_0 ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 5 6 4: − z_post + z_post ≤ 0 ∧ z_post − z_post ≤ 0 ∧ − z_0 + z_0 ≤ 0 ∧ z_0 − z_0 ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − counter_post + counter_post ≤ 0 ∧ counter_post − counter_post ≤ 0 ∧ − counter_0 + counter_0 ≤ 0 ∧ counter_0 − counter_0 ≤ 0

## Proof

The following invariants are asserted.

 0: −127 + y_0 ≤ 0 1: − counter_0 ≤ 0 2: −127 + y_0 ≤ 0 3: counter_post ≤ 0 ∧ − counter_post ≤ 0 ∧ counter_0 ≤ 0 ∧ − counter_0 ≤ 0 4: TRUE 5: TRUE

The invariants are proved as follows.

### IMPACT Invariant Proof

• nodes (location) invariant:  0 (0) −127 + y_0 ≤ 0 1 (1) − counter_0 ≤ 0 2 (2) −127 + y_0 ≤ 0 3 (3) counter_post ≤ 0 ∧ − counter_post ≤ 0 ∧ counter_0 ≤ 0 ∧ − counter_0 ≤ 0 4 (4) TRUE 5 (5) TRUE
• initial node: 5
• cover edges:
• transition edges:  0 0 1 0 1 2 2 4 0 3 2 2 3 3 1 4 5 3 5 6 4

### 2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 2 7 2: − z_post + z_post ≤ 0 ∧ z_post − z_post ≤ 0 ∧ − z_0 + z_0 ≤ 0 ∧ z_0 − z_0 ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − counter_post + counter_post ≤ 0 ∧ counter_post − counter_post ≤ 0 ∧ − counter_0 + counter_0 ≤ 0 ∧ counter_0 − counter_0 ≤ 0
and for every transition t, a duplicate t is considered.

### 3 Transition Removal

We remove transitions 0, 2, 3, 5, 6 using the following ranking functions, which are bounded by −15.

 5: 0 4: 0 3: 0 0: 0 2: 0 1: 0 5: −6 4: −7 3: −8 0: −9 2: −9 2_var_snapshot: −9 2*: −9 1: −13
Hints:
 8 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 1 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 4 lexWeak[ [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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 3 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] ] 5 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] ] 6 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

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

2* 10 2: z_post + z_post ≤ 0z_postz_post ≤ 0z_0 + z_0 ≤ 0z_0z_0 ≤ 0y_0 + y_0 ≤ 0y_0y_0 ≤ 0counter_post + counter_post ≤ 0counter_postcounter_post ≤ 0counter_0 + counter_0 ≤ 0counter_0counter_0 ≤ 0

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

2 8 2_var_snapshot: z_post + z_post ≤ 0z_postz_post ≤ 0z_0 + z_0 ≤ 0z_0z_0 ≤ 0y_0 + y_0 ≤ 0y_0y_0 ≤ 0counter_post + counter_post ≤ 0counter_postcounter_post ≤ 0counter_0 + counter_0 ≤ 0counter_0counter_0 ≤ 0

### 6 SCC Decomposition

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

### 6.1 SCC Subproblem 1/1

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

### 6.1.1 Transition Removal

We remove transition 1 using the following ranking functions, which are bounded by −142.

 0: −1 − 4⋅counter_0 2: 1 − 4⋅counter_0 2_var_snapshot: −4⋅counter_0 2*: 2 − 4⋅counter_0
Hints:
 8 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4] ] 10 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4] ] 1 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 4 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4] ]

### 6.1.2 Transition Removal

We remove transitions 8, 4 using the following ranking functions, which are bounded by −2.

 0: −2 2: 0 2_var_snapshot: −1 2*: 1
Hints:
 8 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 10 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 4 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

### 6.1.3 Transition Removal

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

 0: 0 2: −1 2_var_snapshot: 0 2*: 0
Hints:
 10 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

### 6.1.4 Splitting Cut-Point Transitions

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

### 6.1.4.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 7.

### 6.1.4.1.1 Splitting Cut-Point Transitions

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

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