# LTS Termination Proof

by T2Cert

## Input

Integer Transition System
• Initial Location: 3
• Transitions: (pre-variables and post-variables)  0 0 1: 1 − x_0 ≤ 0 ∧ 1 − y_0 ≤ 0 ∧ − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 1 1 0: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − x_0 + x_post ≤ 0 ∧ −1 + x_0 − x_post ≤ 0 ∧ x_0 − x_post ≤ 0 ∧ − x_0 + x_post ≤ 0 ∧ − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 2 2 1: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − y_0 + y_post ≤ 0 ∧ −1 + y_0 − y_post ≤ 0 ∧ y_0 − y_post ≤ 0 ∧ − y_0 + y_post ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 2 3 0: − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0 3 4 2: − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 0 ∧ − x_0 + x_0 ≤ 0 ∧ x_0 − x_0 ≤ 0

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 0 5 0: − y_post + y_post ≤ 0 ∧ y_post − y_post ≤ 0 ∧ − y_0 + y_0 ≤ 0 ∧ y_0 − y_0 ≤ 0 ∧ − x_post + x_post ≤ 0 ∧ x_post − x_post ≤ 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 2, 3, 4 using the following ranking functions, which are bounded by −11.

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

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

0* 8 0: y_post + y_post ≤ 0y_posty_post ≤ 0y_0 + y_0 ≤ 0y_0y_0 ≤ 0x_post + x_post ≤ 0x_postx_post ≤ 0x_0 + x_0 ≤ 0x_0x_0 ≤ 0

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

0 6 0_var_snapshot: y_post + y_post ≤ 0y_posty_post ≤ 0y_0 + y_0 ≤ 0y_0y_0 ≤ 0x_post + x_post ≤ 0x_postx_post ≤ 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 { 0, 1, 0_var_snapshot, 0* }.

### 5.1.1 Transition Removal

We remove transition 0 using the following ranking functions, which are bounded by 3.

 0: 1 + 3⋅x_0 1: 3⋅x_0 0_var_snapshot: 1 + 3⋅x_0 0*: 2 + 3⋅x_0
Hints:
 6 lexWeak[ [0, 0, 0, 0, 0, 0, 3, 0] ] 8 lexWeak[ [0, 0, 0, 0, 0, 0, 3, 0] ] 0 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 3, 0] , [3, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 1 lexWeak[ [0, 0, 3, 0, 3, 0, 0, 0, 0, 0] ]

### 5.1.2 Transition Removal

We remove transitions 6, 8, 1 using the following ranking functions, which are bounded by −3.

 0: −2 1: 0 0_var_snapshot: −3 0*: −1
Hints:
 6 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0] ] 8 lexStrict[ [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] ]

### 5.1.3 Splitting Cut-Point Transitions

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

### 5.1.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 5.

### 5.1.3.1.1 Splitting Cut-Point Transitions

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

T2Cert

• version: 1.0