# LTS Termination Proof

by T2Cert

## Input

Integer Transition System
• Initial Location: 4
• Transitions: (pre-variables and post-variables)  0 0 1: − i5_0 + length4_0 ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp___08_post + tmp___08_post ≤ 0 ∧ tmp___08_post − tmp___08_post ≤ 0 ∧ − tmp___08_0 + tmp___08_0 ≤ 0 ∧ tmp___08_0 − tmp___08_0 ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − s_post + s_post ≤ 0 ∧ s_post − s_post ≤ 0 ∧ − s_0 + s_0 ≤ 0 ∧ s_0 − s_0 ≤ 0 ∧ − length4_post + length4_post ≤ 0 ∧ length4_post − length4_post ≤ 0 ∧ − length4_0 + length4_0 ≤ 0 ∧ length4_0 − length4_0 ≤ 0 ∧ − i5_post + i5_post ≤ 0 ∧ i5_post − i5_post ≤ 0 ∧ − i5_0 + i5_0 ≤ 0 ∧ i5_0 − i5_0 ≤ 0 0 1 2: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 + i5_0 − length4_0 ≤ 0 ∧ −1 − i5_0 + i5_post ≤ 0 ∧ 1 + i5_0 − i5_post ≤ 0 ∧ i5_0 − i5_post ≤ 0 ∧ − i5_0 + i5_post ≤ 0 ∧ tmp___08_0 − tmp___08_post ≤ 0 ∧ − tmp___08_0 + tmp___08_post ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − s_post + s_post ≤ 0 ∧ s_post − s_post ≤ 0 ∧ − s_0 + s_0 ≤ 0 ∧ s_0 − s_0 ≤ 0 ∧ − length4_post + length4_post ≤ 0 ∧ length4_post − length4_post ≤ 0 ∧ − length4_0 + length4_0 ≤ 0 ∧ length4_0 − length4_0 ≤ 0 2 2 0: − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp___08_post + tmp___08_post ≤ 0 ∧ tmp___08_post − tmp___08_post ≤ 0 ∧ − tmp___08_0 + tmp___08_0 ≤ 0 ∧ tmp___08_0 − tmp___08_0 ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − s_post + s_post ≤ 0 ∧ s_post − s_post ≤ 0 ∧ − s_0 + s_0 ≤ 0 ∧ s_0 − s_0 ≤ 0 ∧ − length4_post + length4_post ≤ 0 ∧ length4_post − length4_post ≤ 0 ∧ − length4_0 + length4_0 ≤ 0 ∧ length4_0 − length4_0 ≤ 0 ∧ − i5_post + i5_post ≤ 0 ∧ i5_post − i5_post ≤ 0 ∧ − i5_0 + i5_0 ≤ 0 ∧ i5_0 − i5_0 ≤ 0 3 3 2: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ s_post − tmp_post ≤ 0 ∧ − s_post + tmp_post ≤ 0 ∧ −10 + length4_post ≤ 0 ∧ 10 − length4_post ≤ 0 ∧ i5_post ≤ 0 ∧ − i5_post ≤ 0 ∧ i5_0 − i5_post ≤ 0 ∧ − i5_0 + i5_post ≤ 0 ∧ length4_0 − length4_post ≤ 0 ∧ − length4_0 + length4_post ≤ 0 ∧ s_0 − s_post ≤ 0 ∧ − s_0 + s_post ≤ 0 ∧ tmp_0 − tmp_post ≤ 0 ∧ − tmp_0 + tmp_post ≤ 0 ∧ − tmp___08_post + tmp___08_post ≤ 0 ∧ tmp___08_post − tmp___08_post ≤ 0 ∧ − tmp___08_0 + tmp___08_0 ≤ 0 ∧ tmp___08_0 − tmp___08_0 ≤ 0 4 4 3: − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp___08_post + tmp___08_post ≤ 0 ∧ tmp___08_post − tmp___08_post ≤ 0 ∧ − tmp___08_0 + tmp___08_0 ≤ 0 ∧ tmp___08_0 − tmp___08_0 ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − s_post + s_post ≤ 0 ∧ s_post − s_post ≤ 0 ∧ − s_0 + s_0 ≤ 0 ∧ s_0 − s_0 ≤ 0 ∧ − length4_post + length4_post ≤ 0 ∧ length4_post − length4_post ≤ 0 ∧ − length4_0 + length4_0 ≤ 0 ∧ length4_0 − length4_0 ≤ 0 ∧ − i5_post + i5_post ≤ 0 ∧ i5_post − i5_post ≤ 0 ∧ − i5_0 + i5_0 ≤ 0 ∧ i5_0 − i5_0 ≤ 0

## Proof

The following invariants are asserted.

 0: −10 + length4_post ≤ 0 ∧ 10 − length4_post ≤ 0 ∧ −10 + length4_0 ≤ 0 ∧ 10 − length4_0 ≤ 0 1: −10 + length4_post ≤ 0 ∧ 10 − length4_post ≤ 0 ∧ 10 − length4_0 ≤ 0 2: −10 + length4_post ≤ 0 ∧ 10 − length4_post ≤ 0 ∧ −10 + length4_0 ≤ 0 ∧ 10 − length4_0 ≤ 0 3: TRUE 4: TRUE

The invariants are proved as follows.

### IMPACT Invariant Proof

• nodes (location) invariant:  0 (0) −10 + length4_post ≤ 0 ∧ 10 − length4_post ≤ 0 ∧ −10 + length4_0 ≤ 0 ∧ 10 − length4_0 ≤ 0 1 (1) −10 + length4_post ≤ 0 ∧ 10 − length4_post ≤ 0 ∧ 10 − length4_0 ≤ 0 2 (2) −10 + length4_post ≤ 0 ∧ 10 − length4_post ≤ 0 ∧ −10 + length4_0 ≤ 0 ∧ 10 − length4_0 ≤ 0 3 (3) TRUE 4 (4) TRUE
• initial node: 4
• cover edges:
• transition edges:  0 0 1 0 1 2 2 2 0 3 3 2 4 4 3

### 2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 2 5 2: − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp___08_post + tmp___08_post ≤ 0 ∧ tmp___08_post − tmp___08_post ≤ 0 ∧ − tmp___08_0 + tmp___08_0 ≤ 0 ∧ tmp___08_0 − tmp___08_0 ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − s_post + s_post ≤ 0 ∧ s_post − s_post ≤ 0 ∧ − s_0 + s_0 ≤ 0 ∧ s_0 − s_0 ≤ 0 ∧ − length4_post + length4_post ≤ 0 ∧ length4_post − length4_post ≤ 0 ∧ − length4_0 + length4_0 ≤ 0 ∧ length4_0 − length4_0 ≤ 0 ∧ − i5_post + i5_post ≤ 0 ∧ i5_post − i5_post ≤ 0 ∧ − i5_0 + i5_0 ≤ 0 ∧ i5_0 − i5_0 ≤ 0
and for every transition t, a duplicate t is considered.

### 3 Transition Removal

We remove transitions 0, 3, 4 using the following ranking functions, which are bounded by −13.

 4: 0 3: 0 0: 0 2: 0 1: 0 4: −5 3: −6 0: −7 2: −7 2_var_snapshot: −7 2*: −7 1: −8
Hints:
 6 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] ] 1 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] ] 2 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 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] ] 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] ] 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] ]

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

2* 8 2: tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp___08_post + tmp___08_post ≤ 0tmp___08_posttmp___08_post ≤ 0tmp___08_0 + tmp___08_0 ≤ 0tmp___08_0tmp___08_0 ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0s_post + s_post ≤ 0s_posts_post ≤ 0s_0 + s_0 ≤ 0s_0s_0 ≤ 0length4_post + length4_post ≤ 0length4_postlength4_post ≤ 0length4_0 + length4_0 ≤ 0length4_0length4_0 ≤ 0i5_post + i5_post ≤ 0i5_posti5_post ≤ 0i5_0 + i5_0 ≤ 0i5_0i5_0 ≤ 0

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

2 6 2_var_snapshot: tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp___08_post + tmp___08_post ≤ 0tmp___08_posttmp___08_post ≤ 0tmp___08_0 + tmp___08_0 ≤ 0tmp___08_0tmp___08_0 ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0s_post + s_post ≤ 0s_posts_post ≤ 0s_0 + s_0 ≤ 0s_0s_0 ≤ 0length4_post + length4_post ≤ 0length4_postlength4_post ≤ 0length4_0 + length4_0 ≤ 0length4_0length4_0 ≤ 0i5_post + i5_post ≤ 0i5_posti5_post ≤ 0i5_0 + i5_0 ≤ 0i5_0i5_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, 2, 2_var_snapshot, 2* }.

### 6.1.1 Transition Removal

We remove transition 1 using the following ranking functions, which are bounded by −128.

 0: −13⋅i5_0 − length4_0 2: 1 − 13⋅i5_0 2_var_snapshot: −13⋅i5_0 2*: 2 − 13⋅i5_0
Hints:
 6 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13] ] 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, 13] ] 1 lexStrict[ [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 13, 0, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 14, 0, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 2 lexWeak[ [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 13] ]

### 6.1.2 Transition Removal

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

 0: − length4_0 − length4_post 2: 0 2_var_snapshot: − length4_0 2*: length4_post
Hints:
 6 lexWeak[ [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0] ] 8 lexStrict[ [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 1, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0] , [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

### 6.1.3 Transition Removal

We remove transition 6 using the following ranking functions, which are bounded by 9.

 0: 0 2: length4_0 2_var_snapshot: 0 2*: 0
Hints:
 6 lexStrict[ [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

### 6.1.4 Splitting Cut-Point Transitions

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

### 6.1.4.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 5.

### 6.1.4.1.1 Splitting Cut-Point Transitions

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

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