# 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 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ 1 − arg2 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ 1 − arg1P ≤ 0 ∧ − arg2P ≤ 0 ∧ arg2P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0 ∧ − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 ∧ − x28 + x28 ≤ 0 ∧ x28 − x28 ≤ 0 ∧ − x19 + x19 ≤ 0 ∧ x19 − x19 ≤ 0 ∧ − x11 + x11 ≤ 0 ∧ x11 − x11 ≤ 0 0 1 1: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − arg2P ≤ 0 ∧ 1 − arg2 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ 1 − arg1P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0 ∧ − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 ∧ − x28 + x28 ≤ 0 ∧ x28 − x28 ≤ 0 ∧ − x19 + x19 ≤ 0 ∧ x19 − x19 ≤ 0 ∧ − x11 + x11 ≤ 0 ∧ x11 − x11 ≤ 0 1 2 2: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ 2 − x7 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ 1 − arg1P ≤ 0 ∧ − arg3P ≤ 0 ∧ arg3P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0 ∧ − x28 + x28 ≤ 0 ∧ x28 − x28 ≤ 0 ∧ − x19 + x19 ≤ 0 ∧ x19 − x19 ≤ 0 ∧ − x11 + x11 ≤ 0 ∧ x11 − x11 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 1 3 2: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 2 − x11 ≤ 0 ∧ − arg3P ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ 1 − arg1P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0 ∧ − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 ∧ − x28 + x28 ≤ 0 ∧ x28 − x28 ≤ 0 ∧ − x19 + x19 ≤ 0 ∧ x19 − x19 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 0 4 3: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ − arg2 ≤ 0 ∧ arg2 ≤ 0 ∧ − arg1P ≤ 0 ∧ arg1P ≤ 0 ∧ − arg2P ≤ 0 ∧ arg2P ≤ 0 ∧ − arg3P ≤ 0 ∧ arg3P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0 ∧ − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 ∧ − x28 + x28 ≤ 0 ∧ x28 − x28 ≤ 0 ∧ − x19 + x19 ≤ 0 ∧ x19 − x19 ≤ 0 ∧ − x11 + x11 ≤ 0 ∧ x11 − x11 ≤ 0 1 5 3: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ − arg1P ≤ 0 ∧ arg1P ≤ 0 ∧ − arg3P ≤ 0 ∧ arg3P ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0 ∧ − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 ∧ − x28 + x28 ≤ 0 ∧ x28 − x28 ≤ 0 ∧ − x19 + x19 ≤ 0 ∧ x19 − x19 ≤ 0 ∧ − x11 + x11 ≤ 0 ∧ x11 − x11 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 2 6 3: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ 2 − x19 ≤ 0 ∧ − arg1P + arg3 ≤ 0 ∧ arg1P − arg3 ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 ∧ − x28 + x28 ≤ 0 ∧ x28 − x28 ≤ 0 ∧ − x11 + x11 ≤ 0 ∧ x11 − x11 ≤ 0 ∧ − arg3P + arg3P ≤ 0 ∧ arg3P − arg3P ≤ 0 ∧ − arg3 + arg3 ≤ 0 ∧ arg3 − arg3 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 3 7 4: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 2 − arg2 ≤ 0 ∧ 2 − arg1 ≤ 0 ∧ 1 − arg2 + x28 ≤ 0 ∧ arg1 − arg2 ≤ 0 ∧ arg1 − arg3 ≤ 0 ∧ − arg1 + arg3 ≤ 0 ∧ arg1 − arg3P ≤ 0 ∧ − arg1 + arg3P ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0 ∧ − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 ∧ − x19 + x19 ≤ 0 ∧ x19 − x19 ≤ 0 ∧ − x11 + x11 ≤ 0 ∧ x11 − x11 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0 4 8 3: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 2 − arg2 ≤ 0 ∧ 2 − arg1 ≤ 0 ∧ arg1 − arg2 ≤ 0 ∧ 1 + arg2P − arg2 ≤ 0 ∧ arg1 − arg3 ≤ 0 ∧ − arg1 + arg3 ≤ 0 ∧ arg1 − arg3P ≤ 0 ∧ − arg1 + arg3P ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0 ∧ − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 ∧ − x28 + x28 ≤ 0 ∧ x28 − x28 ≤ 0 ∧ − x19 + x19 ≤ 0 ∧ x19 − x19 ≤ 0 ∧ − x11 + x11 ≤ 0 ∧ x11 − x11 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0 5 9 0: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ − arg1P + arg1 ≤ 0 ∧ arg1P − arg1 ≤ 0 ∧ − arg2P + arg2 ≤ 0 ∧ arg2P − arg2 ≤ 0 ∧ − arg3P + arg3 ≤ 0 ∧ arg3P − arg3 ≤ 0 ∧ − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 ∧ − x28 + x28 ≤ 0 ∧ x28 − x28 ≤ 0 ∧ − x19 + x19 ≤ 0 ∧ x19 − x19 ≤ 0 ∧ − x11 + x11 ≤ 0 ∧ x11 − x11 ≤ 0

## Proof

The following invariants are asserted.

 0: TRUE 1: 1 − arg1P ≤ 0 ∧ − arg2P ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ − arg2 ≤ 0 2: 1 − arg1P ≤ 0 ∧ − arg2P ≤ 0 ∧ − arg3P ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ − arg2 ≤ 0 ∧ − arg3 ≤ 0 3: TRUE 4: 2 − arg1 ≤ 0 ∧ 2 − arg2 ≤ 0 5: TRUE

The invariants are proved as follows.

### IMPACT Invariant Proof

• nodes (location) invariant:  0 (0) TRUE 1 (1) 1 − arg1P ≤ 0 ∧ − arg2P ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ − arg2 ≤ 0 2 (2) 1 − arg1P ≤ 0 ∧ − arg2P ≤ 0 ∧ − arg3P ≤ 0 ∧ 1 − arg1 ≤ 0 ∧ − arg2 ≤ 0 ∧ − arg3 ≤ 0 3 (3) TRUE 4 (4) 2 − arg1 ≤ 0 ∧ 2 − arg2 ≤ 0 5 (5) TRUE
• initial node: 5
• cover edges:
• transition edges:  0 0 1 0 1 1 0 4 3 1 2 2 1 3 2 1 5 3 2 6 3 3 7 4 4 8 3 5 9 0

### 2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 3 10 3: − x7 + x7 ≤ 0 ∧ x7 − x7 ≤ 0 ∧ − x28 + x28 ≤ 0 ∧ x28 − x28 ≤ 0 ∧ − x19 + x19 ≤ 0 ∧ x19 − x19 ≤ 0 ∧ − x11 + x11 ≤ 0 ∧ x11 − x11 ≤ 0 ∧ − arg3P + arg3P ≤ 0 ∧ arg3P − arg3P ≤ 0 ∧ − arg3 + arg3 ≤ 0 ∧ arg3 − arg3 ≤ 0 ∧ − arg2P + arg2P ≤ 0 ∧ arg2P − arg2P ≤ 0 ∧ − arg2 + arg2 ≤ 0 ∧ arg2 − arg2 ≤ 0 ∧ − arg1P + arg1P ≤ 0 ∧ arg1P − arg1P ≤ 0 ∧ − arg1 + arg1 ≤ 0 ∧ arg1 − arg1 ≤ 0
and for every transition t, a duplicate t is considered.

### 3 Transition Removal

We remove transitions 0, 1, 2, 3, 4, 5, 6, 9 using the following ranking functions, which are bounded by −15.

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

3* 13 3: x7 + x7 ≤ 0x7x7 ≤ 0x28 + x28 ≤ 0x28x28 ≤ 0x19 + x19 ≤ 0x19x19 ≤ 0x11 + x11 ≤ 0x11x11 ≤ 0arg3P + arg3P ≤ 0arg3Parg3P ≤ 0arg3 + arg3 ≤ 0arg3arg3 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

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

3 11 3_var_snapshot: x7 + x7 ≤ 0x7x7 ≤ 0x28 + x28 ≤ 0x28x28 ≤ 0x19 + x19 ≤ 0x19x19 ≤ 0x11 + x11 ≤ 0x11x11 ≤ 0arg3P + arg3P ≤ 0arg3Parg3P ≤ 0arg3 + arg3 ≤ 0arg3arg3 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 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 { 3, 4, 3_var_snapshot, 3* }.

### 6.1.1 Transition Removal

We remove transition 7 using the following ranking functions, which are bounded by 5.

 3: −1 + 4⋅arg2 4: −3 + 4⋅arg2 3_var_snapshot: −2 + 4⋅arg2 3*: 4⋅arg2
Hints:
 11 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0] ] 13 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0] ] 7 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 8 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

### 6.1.2 Transition Removal

We remove transitions 11, 13, 8 using the following ranking functions, which are bounded by −2.

 3: −1 4: 1 3_var_snapshot: −2 3*: 0
Hints:
 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 13 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, 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, 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, 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 10.

### 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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