# LTS Termination Proof

by T2Cert

## Input

Integer Transition System
• Initial Location: 5
• Transitions: (pre-variables and post-variables)  0 0 1: 0 ≤ 0 ∧ 0 ≤ 0 ∧ x_0 ≤ 0 ∧ −1 + x_post ≤ 0 ∧ 1 − x_post ≤ 0 ∧ x_0 − x_post ≤ 0 ∧ − x_0 + x_post ≤ 0 ∧ − x_1 + x_1 ≤ 0 ∧ x_1 − x_1 ≤ 0 0 1 1: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − x_0 ≤ 0 ∧ −1 − x_0 + x_post ≤ 0 ∧ 1 + x_0 − x_post ≤ 0 ∧ x_0 − x_post ≤ 0 ∧ − x_0 + x_post ≤ 0 ∧ − x_1 + x_1 ≤ 0 ∧ x_1 − x_1 ≤ 0 2 2 3: 4 − x_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 2 3 0: −3 + x_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 4 2: 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 4 5 1: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ −5 + x_1 ≤ 0 ∧ 5 − x_1 ≤ 0 ∧ x_0 − x_post ≤ 0 ∧ − x_0 + x_post ≤ 0 5 6 4: 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 Invariant Updates

The following invariants are asserted.

 0: −3 + x_0 ≤ 0 ∧ −5 + x_1 ≤ 0 ∧ 5 − x_1 ≤ 0 1: −5 + x_1 ≤ 0 ∧ 5 − x_1 ≤ 0 2: −5 + x_1 ≤ 0 ∧ 5 − x_1 ≤ 0 3: 4 − x_0 ≤ 0 ∧ −5 + x_1 ≤ 0 ∧ 5 − x_1 ≤ 0 4: TRUE 5: TRUE

The invariants are proved as follows.

### IMPACT Invariant Proof

• nodes (location) invariant:  0 (0) −3 + x_0 ≤ 0 ∧ −5 + x_1 ≤ 0 ∧ 5 − x_1 ≤ 0 1 (1) −5 + x_1 ≤ 0 ∧ 5 − x_1 ≤ 0 2 (2) −5 + x_1 ≤ 0 ∧ 5 − x_1 ≤ 0 3 (3) 4 − x_0 ≤ 0 ∧ −5 + x_1 ≤ 0 ∧ 5 − x_1 ≤ 0 4 (4) TRUE 5 (5) TRUE
• initial node: 5
• cover edges:
• transition edges:  0 0 1 0 1 1 1 4 2 2 2 3 2 3 0 4 5 1 5 6 4

### 2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 1 7 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.

### 3 Transition Removal

We remove transitions 2, 5, 6 using the following ranking functions, which are bounded by −13.

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

### 4 Location Addition

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

1* 10 1: x_post + x_post ≤ 0x_postx_post ≤ 0x_1 + x_1 ≤ 0x_1x_1 ≤ 0x_0 + x_0 ≤ 0x_0x_0 ≤ 0

### 5 Location Addition

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

1 8 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

### 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, 2, 1_var_snapshot, 1* }.

### 6.1.1 Transition Removal

We remove transitions 0, 1 using the following ranking functions, which are bounded by −45.

 0: −13⋅x_0 − x_1 1: −13⋅x_0 + x_1 2: −13⋅x_0 − x_1 1_var_snapshot: −13⋅x_0 1*: 6 − 13⋅x_0
Hints:
 8 lexWeak[ [0, 1, 0, 0, 0, 0, 0, 13] ] 10 lexWeak[ [1, 0, 0, 0, 1, 0, 0, 13] ] 0 lexStrict[ [0, 1, 0, 0, 0, 13, 0, 13, 0, 13, 0, 0] , [0, 1, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0] ] 1 lexStrict[ [0, 1, 0, 0, 0, 0, 0, 13, 0, 13, 0, 0] , [13, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 3 lexWeak[ [0, 0, 0, 0, 0, 0, 1, 0, 13] ] 4 lexWeak[ [0, 1, 0, 0, 0, 0, 0, 1, 0, 13] ]

### 6.1.2 Transition Removal

We remove transitions 8, 10, 3, 4 using the following ranking functions, which are bounded by 4.

 0: 0 1: 15 2: −5 + 2⋅x_1 1_var_snapshot: 2⋅x_1 1*: 4⋅x_1
Hints:
 8 lexStrict[ [2, 0, 0, 0, 2, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0] ] 10 lexStrict[ [0, 4, 0, 0, 0, 0, 0, 0] , [0, 4, 0, 0, 0, 0, 0, 0] ] 3 lexStrict[ [0, 2, 0, 0, 0, 0, 0, 0, 0] , [0, 2, 0, 0, 0, 0, 0, 0, 0] ] 4 lexStrict[ [0, 0, 0, 0, 0, 0, 2, 0, 0, 0] , [0, 2, 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 7.

### 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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