# 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 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ i_14_1 ≤ 0 ∧ − i_14_1 ≤ 0 ∧ − i_14_1 ≤ 0 ∧ i_14_1 ≤ 0 ∧ − i_14_1 ≤ 0 ∧ i_14_1 ≤ 0 ∧ 1 − ___const_10_0 + i_14_1 ≤ 0 ∧ −1 − i_14_1 + i_14_2 ≤ 0 ∧ 1 + i_14_1 − i_14_2 ≤ 0 ∧ 1 − i_14_2 ≤ 0 ∧ −1 + i_14_2 ≤ 0 ∧ 1 − i_14_2 ≤ 0 ∧ −1 + i_14_2 ≤ 0 ∧ 1 − ___const_10_0 + i_14_2 ≤ 0 ∧ −1 − i_14_2 + i_14_post ≤ 0 ∧ 1 + i_14_2 − i_14_post ≤ 0 ∧ 2 − i_14_post ≤ 0 ∧ −2 + i_14_post ≤ 0 ∧ i_14_0 − i_14_post ≤ 0 ∧ − i_14_0 + i_14_post ≤ 0 ∧ − temp0_15_0 + temp0_15_0 ≤ 0 ∧ temp0_15_0 − temp0_15_0 ≤ 0 ∧ − result_12_post + result_12_post ≤ 0 ∧ result_12_post − result_12_post ≤ 0 ∧ − result_12_0 + result_12_0 ≤ 0 ∧ result_12_0 − result_12_0 ≤ 0 ∧ − i_22_0 + i_22_0 ≤ 0 ∧ i_22_0 − i_22_0 ≤ 0 ∧ − ___const_10_0 + ___const_10_0 ≤ 0 ∧ ___const_10_0 − ___const_10_0 ≤ 0 1 1 2: 0 ≤ 0 ∧ 0 ≤ 0 ∧ ___const_10_0 − i_14_0 ≤ 0 ∧ result_12_post − temp0_15_0 ≤ 0 ∧ − result_12_post + temp0_15_0 ≤ 0 ∧ result_12_0 − result_12_post ≤ 0 ∧ − result_12_0 + result_12_post ≤ 0 ∧ − temp0_15_0 + temp0_15_0 ≤ 0 ∧ temp0_15_0 − temp0_15_0 ≤ 0 ∧ − i_22_0 + i_22_0 ≤ 0 ∧ i_22_0 − i_22_0 ≤ 0 ∧ − i_14_post + i_14_post ≤ 0 ∧ i_14_post − i_14_post ≤ 0 ∧ − i_14_2 + i_14_2 ≤ 0 ∧ i_14_2 − i_14_2 ≤ 0 ∧ − i_14_1 + i_14_1 ≤ 0 ∧ i_14_1 − i_14_1 ≤ 0 ∧ − i_14_0 + i_14_0 ≤ 0 ∧ i_14_0 − i_14_0 ≤ 0 ∧ − ___const_10_0 + ___const_10_0 ≤ 0 ∧ ___const_10_0 − ___const_10_0 ≤ 0 1 2 3: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − ___const_10_0 + i_14_0 ≤ 0 ∧ −1 − i_14_0 + i_14_post ≤ 0 ∧ 1 + i_14_0 − i_14_post ≤ 0 ∧ −1 + i_14_post − i_22_0 ≤ 0 ∧ 1 − i_14_post + i_22_0 ≤ 0 ∧ 1 − ___const_10_0 + i_22_0 ≤ 0 ∧ i_14_0 − i_14_post ≤ 0 ∧ − i_14_0 + i_14_post ≤ 0 ∧ − temp0_15_0 + temp0_15_0 ≤ 0 ∧ temp0_15_0 − temp0_15_0 ≤ 0 ∧ − result_12_post + result_12_post ≤ 0 ∧ result_12_post − result_12_post ≤ 0 ∧ − result_12_0 + result_12_0 ≤ 0 ∧ result_12_0 − result_12_0 ≤ 0 ∧ − i_22_0 + i_22_0 ≤ 0 ∧ i_22_0 − i_22_0 ≤ 0 ∧ − i_14_2 + i_14_2 ≤ 0 ∧ i_14_2 − i_14_2 ≤ 0 ∧ − i_14_1 + i_14_1 ≤ 0 ∧ i_14_1 − i_14_1 ≤ 0 ∧ − ___const_10_0 + ___const_10_0 ≤ 0 ∧ ___const_10_0 − ___const_10_0 ≤ 0 3 3 1: − temp0_15_0 + temp0_15_0 ≤ 0 ∧ temp0_15_0 − temp0_15_0 ≤ 0 ∧ − result_12_post + result_12_post ≤ 0 ∧ result_12_post − result_12_post ≤ 0 ∧ − result_12_0 + result_12_0 ≤ 0 ∧ result_12_0 − result_12_0 ≤ 0 ∧ − i_22_0 + i_22_0 ≤ 0 ∧ i_22_0 − i_22_0 ≤ 0 ∧ − i_14_post + i_14_post ≤ 0 ∧ i_14_post − i_14_post ≤ 0 ∧ − i_14_2 + i_14_2 ≤ 0 ∧ i_14_2 − i_14_2 ≤ 0 ∧ − i_14_1 + i_14_1 ≤ 0 ∧ i_14_1 − i_14_1 ≤ 0 ∧ − i_14_0 + i_14_0 ≤ 0 ∧ i_14_0 − i_14_0 ≤ 0 ∧ − ___const_10_0 + ___const_10_0 ≤ 0 ∧ ___const_10_0 − ___const_10_0 ≤ 0 4 4 0: − temp0_15_0 + temp0_15_0 ≤ 0 ∧ temp0_15_0 − temp0_15_0 ≤ 0 ∧ − result_12_post + result_12_post ≤ 0 ∧ result_12_post − result_12_post ≤ 0 ∧ − result_12_0 + result_12_0 ≤ 0 ∧ result_12_0 − result_12_0 ≤ 0 ∧ − i_22_0 + i_22_0 ≤ 0 ∧ i_22_0 − i_22_0 ≤ 0 ∧ − i_14_post + i_14_post ≤ 0 ∧ i_14_post − i_14_post ≤ 0 ∧ − i_14_2 + i_14_2 ≤ 0 ∧ i_14_2 − i_14_2 ≤ 0 ∧ − i_14_1 + i_14_1 ≤ 0 ∧ i_14_1 − i_14_1 ≤ 0 ∧ − i_14_0 + i_14_0 ≤ 0 ∧ i_14_0 − i_14_0 ≤ 0 ∧ − ___const_10_0 + ___const_10_0 ≤ 0 ∧ ___const_10_0 − ___const_10_0 ≤ 0

## Proof

The following invariants are asserted.

 0: TRUE 1: i_14_1 ≤ 0 ∧ − i_14_1 ≤ 0 ∧ −1 + i_14_2 ≤ 0 ∧ 1 − i_14_2 ≤ 0 2: i_14_1 ≤ 0 ∧ − i_14_1 ≤ 0 ∧ −1 + i_14_2 ≤ 0 ∧ 1 − i_14_2 ≤ 0 3: i_14_1 ≤ 0 ∧ − i_14_1 ≤ 0 ∧ −1 + i_14_2 ≤ 0 ∧ 1 − i_14_2 ≤ 0 4: TRUE

The invariants are proved as follows.

### IMPACT Invariant Proof

• nodes (location) invariant:  0 (0) TRUE 1 (1) i_14_1 ≤ 0 ∧ − i_14_1 ≤ 0 ∧ −1 + i_14_2 ≤ 0 ∧ 1 − i_14_2 ≤ 0 2 (2) i_14_1 ≤ 0 ∧ − i_14_1 ≤ 0 ∧ −1 + i_14_2 ≤ 0 ∧ 1 − i_14_2 ≤ 0 3 (3) i_14_1 ≤ 0 ∧ − i_14_1 ≤ 0 ∧ −1 + i_14_2 ≤ 0 ∧ 1 − i_14_2 ≤ 0 4 (4) TRUE
• initial node: 4
• cover edges:
• transition edges:  0 0 1 1 1 2 1 2 3 3 3 1 4 4 0

### 2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 1 5 1: − temp0_15_0 + temp0_15_0 ≤ 0 ∧ temp0_15_0 − temp0_15_0 ≤ 0 ∧ − result_12_post + result_12_post ≤ 0 ∧ result_12_post − result_12_post ≤ 0 ∧ − result_12_0 + result_12_0 ≤ 0 ∧ result_12_0 − result_12_0 ≤ 0 ∧ − i_22_0 + i_22_0 ≤ 0 ∧ i_22_0 − i_22_0 ≤ 0 ∧ − i_14_post + i_14_post ≤ 0 ∧ i_14_post − i_14_post ≤ 0 ∧ − i_14_2 + i_14_2 ≤ 0 ∧ i_14_2 − i_14_2 ≤ 0 ∧ − i_14_1 + i_14_1 ≤ 0 ∧ i_14_1 − i_14_1 ≤ 0 ∧ − i_14_0 + i_14_0 ≤ 0 ∧ i_14_0 − i_14_0 ≤ 0 ∧ − ___const_10_0 + ___const_10_0 ≤ 0 ∧ ___const_10_0 − ___const_10_0 ≤ 0
and for every transition t, a duplicate t is considered.

### 3 Transition Removal

We remove transitions 0, 1, 4 using the following ranking functions, which are bounded by −13.

 4: 0 0: 0 1: 0 3: 0 2: 0 4: −5 0: −6 1: −7 3: −7 1_var_snapshot: −7 1*: −7 2: −11
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] ] 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, 0, 0, 0] ] 3 lexWeak[ [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, 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] ] 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] ]

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

1* 8 1: temp0_15_0 + temp0_15_0 ≤ 0temp0_15_0temp0_15_0 ≤ 0result_12_post + result_12_post ≤ 0result_12_postresult_12_post ≤ 0result_12_0 + result_12_0 ≤ 0result_12_0result_12_0 ≤ 0i_22_0 + i_22_0 ≤ 0i_22_0i_22_0 ≤ 0i_14_post + i_14_post ≤ 0i_14_posti_14_post ≤ 0i_14_2 + i_14_2 ≤ 0i_14_2i_14_2 ≤ 0i_14_1 + i_14_1 ≤ 0i_14_1i_14_1 ≤ 0i_14_0 + i_14_0 ≤ 0i_14_0i_14_0 ≤ 0___const_10_0 + ___const_10_0 ≤ 0___const_10_0___const_10_0 ≤ 0

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

1 6 1_var_snapshot: temp0_15_0 + temp0_15_0 ≤ 0temp0_15_0temp0_15_0 ≤ 0result_12_post + result_12_post ≤ 0result_12_postresult_12_post ≤ 0result_12_0 + result_12_0 ≤ 0result_12_0result_12_0 ≤ 0i_22_0 + i_22_0 ≤ 0i_22_0i_22_0 ≤ 0i_14_post + i_14_post ≤ 0i_14_posti_14_post ≤ 0i_14_2 + i_14_2 ≤ 0i_14_2i_14_2 ≤ 0i_14_1 + i_14_1 ≤ 0i_14_1i_14_1 ≤ 0i_14_0 + i_14_0 ≤ 0i_14_0i_14_0 ≤ 0___const_10_0 + ___const_10_0 ≤ 0___const_10_0___const_10_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 { 1, 3, 1_var_snapshot, 1* }.

### 6.1.1 Transition Removal

We remove transition 2 using the following ranking functions, which are bounded by −4.

 1: −1 − 4⋅i_14_0 − i_14_2 + 4⋅i_22_0 3: −4⋅i_14_0 + 4⋅i_22_0 1_var_snapshot: −3 − 4⋅i_14_0 + 4⋅i_22_0 1*: −4⋅i_14_0 − i_14_2 + 4⋅i_22_0
Hints:
 6 lexWeak[ [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0] ] 8 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0] ] 2 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 3 lexWeak[ [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0] ]

### 6.1.2 Transition Removal

We remove transitions 8, 3 using the following ranking functions, which are bounded by 0.

 1: 0 3: 2 1_var_snapshot: 0 1*: 1
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] ] 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] ] 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] ]

### 6.1.3 Transition Removal

We remove transition 6 using the following ranking functions, which are bounded by 0.

 1: 1 3: 0 1_var_snapshot: 0 1*: 0
Hints:
 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] ]

### 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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