# LTS Termination Proof

by T2Cert

## Input

Integer Transition System
• Initial Location: 6
• Transitions: (pre-variables and post-variables)  0 0 1: 1 ≤ 0 ∧ − b_post + b_post ≤ 0 ∧ b_post − b_post ≤ 0 ∧ − b_0 + b_0 ≤ 0 ∧ b_0 − b_0 ≤ 0 ∧ − a_post + a_post ≤ 0 ∧ a_post − a_post ≤ 0 ∧ − a_0 + a_0 ≤ 0 ∧ a_0 − a_0 ≤ 0 1 1 0: − b_post + b_post ≤ 0 ∧ b_post − b_post ≤ 0 ∧ − b_0 + b_0 ≤ 0 ∧ b_0 − b_0 ≤ 0 ∧ − a_post + a_post ≤ 0 ∧ a_post − a_post ≤ 0 ∧ − a_0 + a_0 ≤ 0 ∧ a_0 − a_0 ≤ 0 0 2 2: 0 ≤ 0 ∧ 0 ≤ 0 ∧ b_post ≤ 0 ∧ − b_post ≤ 0 ∧ b_0 − b_post ≤ 0 ∧ − b_0 + b_post ≤ 0 ∧ − a_post + a_post ≤ 0 ∧ a_post − a_post ≤ 0 ∧ − a_0 + a_0 ≤ 0 ∧ a_0 − a_0 ≤ 0 2 3 3: b_0 ≤ 0 ∧ − b_post + b_post ≤ 0 ∧ b_post − b_post ≤ 0 ∧ − b_0 + b_0 ≤ 0 ∧ b_0 − b_0 ≤ 0 ∧ − a_post + a_post ≤ 0 ∧ a_post − a_post ≤ 0 ∧ − a_0 + a_0 ≤ 0 ∧ a_0 − a_0 ≤ 0 4 4 0: a_0 ≤ 0 ∧ − b_post + b_post ≤ 0 ∧ b_post − b_post ≤ 0 ∧ − b_0 + b_0 ≤ 0 ∧ b_0 − b_0 ≤ 0 ∧ − a_post + a_post ≤ 0 ∧ a_post − a_post ≤ 0 ∧ − a_0 + a_0 ≤ 0 ∧ a_0 − a_0 ≤ 0 4 5 2: 0 ≤ 0 ∧ 0 ≤ 0 ∧ −1 + b_post ≤ 0 ∧ 1 − b_post ≤ 0 ∧ b_0 − b_post ≤ 0 ∧ − b_0 + b_post ≤ 0 ∧ − a_post + a_post ≤ 0 ∧ a_post − a_post ≤ 0 ∧ − a_0 + a_0 ≤ 0 ∧ a_0 − a_0 ≤ 0 5 6 4: 0 ≤ 0 ∧ 0 ≤ 0 ∧ −1 + a_post ≤ 0 ∧ 1 − a_post ≤ 0 ∧ a_0 − a_post ≤ 0 ∧ − a_0 + a_post ≤ 0 ∧ − b_post + b_post ≤ 0 ∧ b_post − b_post ≤ 0 ∧ − b_0 + b_0 ≤ 0 ∧ b_0 − b_0 ≤ 0 6 7 5: − b_post + b_post ≤ 0 ∧ b_post − b_post ≤ 0 ∧ − b_0 + b_0 ≤ 0 ∧ b_0 − b_0 ≤ 0 ∧ − a_post + a_post ≤ 0 ∧ a_post − a_post ≤ 0 ∧ − a_0 + a_0 ≤ 0 ∧ a_0 − a_0 ≤ 0

## Proof

### 1 Invariant Updates

The following invariants are asserted.

 0: −1 + a_post ≤ 0 ∧ 1 − a_post ≤ 0 ∧ 1 − a_0 ≤ 0 ∧ a_0 ≤ 0 1: −1 + a_post ≤ 0 ∧ 1 − a_post ≤ 0 ∧ 1 − a_0 ≤ 0 ∧ a_0 ≤ 0 2: −1 + a_post ≤ 0 ∧ 1 − a_post ≤ 0 ∧ −1 + a_0 ≤ 0 ∧ 1 − a_0 ≤ 0 ∧ −1 + b_post ≤ 0 ∧ −1 + b_0 ≤ 0 3: −1 + a_post ≤ 0 ∧ 1 − a_post ≤ 0 ∧ −1 + a_0 ≤ 0 ∧ 1 − a_0 ≤ 0 ∧ −1 + b_post ≤ 0 ∧ b_0 ≤ 0 4: −1 + a_post ≤ 0 ∧ 1 − a_post ≤ 0 ∧ −1 + a_0 ≤ 0 ∧ 1 − a_0 ≤ 0 5: TRUE 6: TRUE

The invariants are proved as follows.

### IMPACT Invariant Proof

• nodes (location) invariant:  0 (0) −1 + a_post ≤ 0 ∧ 1 − a_post ≤ 0 ∧ 1 − a_0 ≤ 0 ∧ a_0 ≤ 0 1 (1) −1 + a_post ≤ 0 ∧ 1 − a_post ≤ 0 ∧ 1 − a_0 ≤ 0 ∧ a_0 ≤ 0 2 (2) −1 + a_post ≤ 0 ∧ 1 − a_post ≤ 0 ∧ −1 + a_0 ≤ 0 ∧ 1 − a_0 ≤ 0 ∧ −1 + b_post ≤ 0 ∧ −1 + b_0 ≤ 0 3 (3) −1 + a_post ≤ 0 ∧ 1 − a_post ≤ 0 ∧ −1 + a_0 ≤ 0 ∧ 1 − a_0 ≤ 0 ∧ −1 + b_post ≤ 0 ∧ b_0 ≤ 0 4 (4) −1 + a_post ≤ 0 ∧ 1 − a_post ≤ 0 ∧ −1 + a_0 ≤ 0 ∧ 1 − a_0 ≤ 0 5 (5) TRUE 6 (6) TRUE
• initial node: 6
• cover edges:
• transition edges:  0 0 1 0 2 2 1 1 0 2 3 3 4 4 0 4 5 2 5 6 4 6 7 5

### 2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 0 8 0: − b_post + b_post ≤ 0 ∧ b_post − b_post ≤ 0 ∧ − b_0 + b_0 ≤ 0 ∧ b_0 − b_0 ≤ 0 ∧ − a_post + a_post ≤ 0 ∧ a_post − a_post ≤ 0 ∧ − a_0 + a_0 ≤ 0 ∧ a_0 − a_0 ≤ 0
and for every transition t, a duplicate t is considered.

### 3 Transition Removal

We remove transitions 2, 3, 4, 5, 6, 7 using the following ranking functions, which are bounded by −17.

 6: 0 5: 0 4: 0 0: 0 1: 0 2: 0 3: 0 6: −7 5: −8 4: −9 0: −10 1: −10 0_var_snapshot: −10 0*: −10 2: −14 3: −15
Hints:
 9 lexWeak[ [0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0] ] 0 lexWeak[ [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0] ] 1 lexWeak[ [0, 0, 1, 1, 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] ] 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] ] 4 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] ] 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, 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] ] 7 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0] ]

### 4 Location Addition

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

0* 11 0: b_post + b_post ≤ 0b_postb_post ≤ 0b_0 + b_0 ≤ 0b_0b_0 ≤ 0a_post + a_post ≤ 0a_posta_post ≤ 0a_0 + a_0 ≤ 0a_0a_0 ≤ 0

### 5 Location Addition

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

0 9 0_var_snapshot: b_post + b_post ≤ 0b_postb_post ≤ 0b_0 + b_0 ≤ 0b_0b_0 ≤ 0a_post + a_post ≤ 0a_posta_post ≤ 0a_0 + a_0 ≤ 0a_0a_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, 1, 0_var_snapshot, 0* }.

### 6.1.1 Transition Removal

We remove transitions 11, 0, 1 using the following ranking functions, which are bounded by −3.

 0: −2 1: 0 0_var_snapshot: 0 0*: −1
Hints:
 9 lexWeak[ [0, 0, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0] ] 11 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 lexStrict[ [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 1 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] ]

### 6.1.2 Transition Removal

We remove transition 9 using the following ranking functions, which are bounded by −1.

 0: 0 1: 0 0_var_snapshot: 0 0*: 0
Hints:
 9 lexStrict[ [0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

### 6.1.3 Splitting Cut-Point Transitions

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

### 6.1.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 8.

### 6.1.3.1.1 Splitting Cut-Point Transitions

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

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