# LTS Termination Proof

by T2Cert

## Input

Integer Transition System
• Initial Location: 4
• Transitions: (pre-variables and post-variables)  0 0 1: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ y_0 ≤ 0 ∧ 1 − x_0 + x_post ≤ 0 ∧ −1 + x_0 − x_post ≤ 0 ∧ −1 − y_0 + y_post ≤ 0 ∧ 1 + y_0 − y_post ≤ 0 ∧ x_0 − x_post ≤ 0 ∧ − x_0 + x_post ≤ 0 ∧ y_0 − y_post ≤ 0 ∧ − y_0 + y_post ≤ 0 0 1 2: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − y_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 2 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 3 1: − 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 4 0: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − x_0 ≤ 0 ∧ − x_0 + y_post ≤ 0 ∧ x_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 4 5 3: − 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

The following invariants are asserted.

 0: 1 − x_0 ≤ 0 1: TRUE 2: 1 − x_0 ≤ 0 3: TRUE 4: TRUE

The invariants are proved as follows.

### IMPACT Invariant Proof

• nodes (location) invariant:  0 (0) 1 − x_0 ≤ 0 1 (1) TRUE 2 (2) 1 − x_0 ≤ 0 3 (3) TRUE 4 (4) TRUE
• initial node: 4
• cover edges:
• transition edges:  0 0 1 0 1 2 1 4 0 2 2 0 3 3 1 4 5 3

### 2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 0 6 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 13 1: − 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.

### 3 Transition Removal

We remove transitions 3, 5 using the following ranking functions, which are bounded by −13.

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

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

1* 16 1: 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.

1 14 1_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

### 8 SCC Decomposition

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

### 8.1 SCC Subproblem 1/1

Here we consider the SCC { 0, 2, 1, 0_var_snapshot, 0*, 1_var_snapshot, 1* }.

### 8.1.1 Transition Removal

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

 0: 4⋅x_0 2: 4⋅x_0 1: 2 + 4⋅x_0 0_var_snapshot: 4⋅x_0 0*: 4⋅x_0 1_var_snapshot: 1 + 4⋅x_0 1*: 3 + 4⋅x_0
Hints:
 7 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 4, 0] ] 9 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 4, 0] ] 14 lexWeak[ [0, 0, 0, 0, 0, 0, 4, 0] ] 16 lexWeak[ [0, 0, 0, 0, 0, 0, 4, 0] ] 0 lexStrict[ [0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0] , [4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 1 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0] ] 2 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 4, 0] ] 4 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0] ]

### 8.1.2 Transition Removal

We remove transitions 14, 4 using the following ranking functions, which are bounded by 2.

 0: 1 − 4⋅x_0 + 4⋅y_0 2: 3 − 4⋅x_0 + 4⋅y_0 1: 4 0_var_snapshot: −4⋅x_0 + 4⋅y_0 0*: 2 − 4⋅x_0 + 4⋅y_0 1_var_snapshot: 3 1*: 4
Hints:
 7 lexWeak[ [0, 0, 0, 4, 0, 0, 0, 0, 4] ] 9 lexWeak[ [0, 0, 0, 4, 0, 0, 0, 0, 4] ] 14 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0] ] 16 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0] ] 1 lexWeak[ [0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 4] ] 2 lexWeak[ [0, 0, 0, 4, 0, 0, 0, 0, 4] ] 4 lexStrict[ [0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 4] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

### 8.1.3 Transition Removal

We remove transitions 16, 1 using the following ranking functions, which are bounded by 0.

 0: −2 + 4⋅y_0 2: 4⋅y_0 1: 0 0_var_snapshot: −3 + 4⋅y_0 0*: −1 + 4⋅y_0 1_var_snapshot: 0 1*: 1
Hints:
 7 lexWeak[ [0, 0, 0, 4, 0, 0, 0, 0, 0] ] 9 lexWeak[ [0, 0, 0, 4, 0, 0, 0, 0, 0] ] 16 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0] ] 1 lexStrict[ [0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 0] , [0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0] ] 2 lexWeak[ [0, 0, 0, 4, 0, 0, 0, 0, 0] ]

### 8.1.4 Transition Removal

We remove transitions 7, 9, 2 using the following ranking functions, which are bounded by −1.

 0: 0 2: 2⋅x_0 1: 0 0_var_snapshot: −1 0*: x_0 1_var_snapshot: 0 1*: 0
Hints:
 7 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0] ] 9 lexStrict[ [1, 0, 0, 0, 0, 0, 0, 0, 0] , [1, 0, 0, 0, 0, 0, 0, 0, 0] ] 2 lexStrict[ [1, 0, 0, 0, 0, 0, 0, 1, 0] , [2, 0, 0, 0, 0, 0, 0, 0, 0] ]

### 8.1.5 Splitting Cut-Point Transitions

We consider 2 subproblems corresponding to sets of cut-point transitions as follows.

### 8.1.5.1 Cut-Point Subproblem 1/2

Here we consider cut-point transition 6.

### 8.1.5.1.1 Splitting Cut-Point Transitions

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

### 8.1.5.2 Cut-Point Subproblem 2/2

Here we consider cut-point transition 13.

### 8.1.5.2.1 Splitting Cut-Point Transitions

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

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