# LTS Termination Proof

by T2Cert

## Input

Integer Transition System
• Initial Location: 3
• Transitions: (pre-variables and post-variables)  0 0 1: 0 ≤ 0 ∧ 0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_1 + x_1 ≤ 0 ∧ x_1 − x_1 ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 1 1 2: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − x_0 ≤ 0 ∧ − x_0 + x_1 ≤ 0 ∧ x_0 − x_1 ≤ 0 ∧ 1 − x_1 + x_post ≤ 0 ∧ −1 + x_1 − x_post ≤ 0 ∧ x_0 − x_post ≤ 0 ∧ − x_0 + x_post ≤ 0 2 2 1: 0 ≤ 0 ∧ 0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_1 + x_1 ≤ 0 ∧ x_1 − x_1 ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 3 3 0: 0 ≤ 0 ∧ 0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_1 + x_1 ≤ 0 ∧ x_1 − x_1 ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 1 4 1: − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_1 + x_1 ≤ 0 ∧ x_1 − x_1 ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0
and for every transition t, a duplicate t is considered.

### 2 Transition Removal

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

 3: 0 0: 0 1: 0 2: 0 3: −4 0: −5 1: −6 2: −6 1_var_snapshot: −6 1*: −6
Hints:
 5 lexWeak[ [0, 0, 0, 0, 0, 0] ] 1 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 2 lexWeak[ [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] ] 3 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* 7 1: x_post + x_post ≤ 0x_postx_post ≤ 0x_1 + x_1 ≤ 0x_1x_1 ≤ 0x_0 + x_0 ≤ 0x_0x_0 ≤ 0

### 4 Location Addition

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

1 5 1_var_snapshot: x_post + x_post ≤ 0x_postx_post ≤ 0x_1 + x_1 ≤ 0x_1x_1 ≤ 0x_0 + x_0 ≤ 0x_0x_0 ≤ 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, 2, 1_var_snapshot, 1* }.

### 5.1.1 Transition Removal

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

 1: −2 + 4⋅x_0 2: 4⋅x_0 1_var_snapshot: −2 + 4⋅x_0 1*: −1 + 4⋅x_0
Hints:
 5 lexWeak[ [0, 0, 0, 0, 4, 0] ] 7 lexWeak[ [0, 0, 0, 0, 4, 0] ] 1 lexStrict[ [0, 0, 0, 0, 0, 4, 0, 4, 0, 4, 0] , [0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0] ] 2 lexWeak[ [0, 0, 0, 0, 0, 0, 4, 0] ]

### 5.1.2 Transition Removal

We remove transitions 5, 2 using the following ranking functions, which are bounded by −1.

 1: 0 2: 2 1_var_snapshot: −1 1*: 1
Hints:
 5 lexStrict[ [0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0] ] 7 lexWeak[ [0, 0, 0, 0, 0, 0] ] 2 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0] ]

### 5.1.3 Transition Removal

We remove transition 7 using the following ranking functions, which are bounded by −1.

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

### 5.1.4 Splitting Cut-Point Transitions

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

### 5.1.4.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 4.

### 5.1.4.1.1 Splitting Cut-Point Transitions

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

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