LTS Termination Proof

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Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

0: TRUE
1: 1 − arg1P ≤ 01 − arg1 ≤ 0
2: arg1P ≤ 0arg1 ≤ 0
3: TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
1 5 1: arg3P + arg3P ≤ 0arg3Parg3P ≤ 0arg3 + arg3 ≤ 0arg3arg3 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0
2 12 2: arg3P + arg3P ≤ 0arg3Parg3P ≤ 0arg3 + arg3 ≤ 0arg3arg3 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0
and for every transition t, a duplicate t is considered.

3 Transition Removal

We remove transitions 0, 2, 4 using the following ranking functions, which are bounded by −15.

3: 0
0: 0
1: 0
2: 0
3: −5
0: −6
1: −7
1_var_snapshot: −7
1*: −7
2: −10
2_var_snapshot: −10
2*: −10
Hints:
6 lexWeak[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
13 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, 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 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] ]
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] ]
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] ]

4 Location Addition

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

1* 8 1: arg3P + arg3P ≤ 0arg3Parg3P ≤ 0arg3 + arg3 ≤ 0arg3arg3 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

5 Location Addition

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

1 6 1_var_snapshot: arg3P + arg3P ≤ 0arg3Parg3P ≤ 0arg3 + arg3 ≤ 0arg3arg3 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

6 Location Addition

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

2* 15 2: arg3P + arg3P ≤ 0arg3Parg3P ≤ 0arg3 + arg3 ≤ 0arg3arg3 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

7 Location Addition

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

2 13 2_var_snapshot: arg3P + arg3P ≤ 0arg3Parg3P ≤ 0arg3 + arg3 ≤ 0arg3arg3 ≤ 0arg2P + arg2P ≤ 0arg2Parg2P ≤ 0arg2 + arg2 ≤ 0arg2arg2 ≤ 0arg1P + arg1P ≤ 0arg1Parg1P ≤ 0arg1 + arg1 ≤ 0arg1arg1 ≤ 0

8 SCC Decomposition

We consider subproblems for each of the 2 SCC(s) of the program graph.

8.1 SCC Subproblem 1/2

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

8.1.1 Transition Removal

We remove transitions 13, 15, 3 using the following ranking functions, which are bounded by −1.

2: 1 + 3⋅arg1
2_var_snapshot: 3⋅arg1
2*: 2 + 3⋅arg1
Hints:
13 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0] , [0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
15 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0] , [0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
3 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 3, 0, 0, 0, 0, 0] , [0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

8.1.2 Splitting Cut-Point Transitions

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

8.1.2.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 12.

8.1.2.1.1 Splitting Cut-Point Transitions

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

8.2 SCC Subproblem 2/2

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

8.2.1 Splitting Cut-Point Transitions

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

8.2.1.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 5.

8.2.1.1.1 Fresh Variable Addition

The new variable __snapshot_1_arg3P is introduced. The transition formulas are extended as follows:

6: __snapshot_1_arg3Parg3Parg3P__snapshot_1_arg3P
8: __snapshot_1_arg3P__snapshot_1_arg3P__snapshot_1_arg3P__snapshot_1_arg3P
1: __snapshot_1_arg3P__snapshot_1_arg3P__snapshot_1_arg3P__snapshot_1_arg3P

8.2.1.1.2 Fresh Variable Addition

The new variable __snapshot_1_arg3 is introduced. The transition formulas are extended as follows:

6: __snapshot_1_arg3arg3arg3__snapshot_1_arg3
8: __snapshot_1_arg3__snapshot_1_arg3__snapshot_1_arg3__snapshot_1_arg3
1: __snapshot_1_arg3__snapshot_1_arg3__snapshot_1_arg3__snapshot_1_arg3

8.2.1.1.3 Fresh Variable Addition

The new variable __snapshot_1_arg2P is introduced. The transition formulas are extended as follows:

6: __snapshot_1_arg2Parg2Parg2P__snapshot_1_arg2P
8: __snapshot_1_arg2P__snapshot_1_arg2P__snapshot_1_arg2P__snapshot_1_arg2P
1: __snapshot_1_arg2P__snapshot_1_arg2P__snapshot_1_arg2P__snapshot_1_arg2P

8.2.1.1.4 Fresh Variable Addition

The new variable __snapshot_1_arg2 is introduced. The transition formulas are extended as follows:

6: __snapshot_1_arg2arg2arg2__snapshot_1_arg2
8: __snapshot_1_arg2__snapshot_1_arg2__snapshot_1_arg2__snapshot_1_arg2
1: __snapshot_1_arg2__snapshot_1_arg2__snapshot_1_arg2__snapshot_1_arg2

8.2.1.1.5 Fresh Variable Addition

The new variable __snapshot_1_arg1P is introduced. The transition formulas are extended as follows:

6: __snapshot_1_arg1Parg1Parg1P__snapshot_1_arg1P
8: __snapshot_1_arg1P__snapshot_1_arg1P__snapshot_1_arg1P__snapshot_1_arg1P
1: __snapshot_1_arg1P__snapshot_1_arg1P__snapshot_1_arg1P__snapshot_1_arg1P

8.2.1.1.6 Fresh Variable Addition

The new variable __snapshot_1_arg1 is introduced. The transition formulas are extended as follows:

6: __snapshot_1_arg1arg1arg1__snapshot_1_arg1
8: __snapshot_1_arg1__snapshot_1_arg1__snapshot_1_arg1__snapshot_1_arg1
1: __snapshot_1_arg1__snapshot_1_arg1__snapshot_1_arg1__snapshot_1_arg1

8.2.1.1.7 Invariant Updates

The following invariants are asserted.

0: TRUE
1: 1 + arg2arg3P ≤ 0arg3P ≤ 01 − arg1P ≤ 01 − arg1 ≤ 0−1 + arg3arg3P ≤ 01 + arg2arg3P ≤ 01 − arg1P ≤ 01 − arg1 ≤ 0
2: arg1P ≤ 0arg1 ≤ 0
3: TRUE
1: 1 + arg2arg3P ≤ 0arg3P ≤ 01 − arg1P ≤ 01 − arg1 ≤ 0−1 + arg3arg3P ≤ 01 − __snapshot_1_arg3P + arg3P ≤ 01 + arg2arg3P ≤ 0__snapshot_1_arg3P ≤ 01 − arg1P ≤ 01 − arg1 ≤ 0−1 + arg3arg3P ≤ 01 + arg2arg3P ≤ 01 − arg1P ≤ 01 − arg1 ≤ 0
1_var_snapshot: 1 − __snapshot_1_arg3P + arg2 ≤ 0__snapshot_1_arg3P ≤ 01 − arg1P ≤ 01 − arg1 ≤ 01 − __snapshot_1_arg3P + arg2 ≤ 0−1 − __snapshot_1_arg3P + arg3 ≤ 01 − arg1P ≤ 01 − arg1 ≤ 0
1*: −1 + arg3arg3P ≤ 01 − __snapshot_1_arg3P + arg3P ≤ 01 + arg2arg3P ≤ 0__snapshot_1_arg3P ≤ 01 − arg1P ≤ 01 − arg1 ≤ 0

The invariants are proved as follows.

IMPACT Invariant Proof

8.2.1.1.8 Transition Removal

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

1: arg3P
1_var_snapshot: __snapshot_1_arg3P
1*: __snapshot_1_arg3P
Hints:
6 distribute assertion
lexWeak[ [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
lexWeak[ [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
lexWeak[ [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
8 lexStrict[ [0, 1, 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, 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, 0, 0, 0, 0] ]
1 distribute assertion
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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]
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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

8.2.1.1.9 Transition Removal

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

1: −1
1_var_snapshot: −2
1*: −3
Hints:
6 distribute assertion
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] ]
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] ]
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] ]
1 distribute assertion
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, 0, 0, 0, 0, 0] ]
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, 0, 0, 0, 0, 0] ]

8.2.1.1.10 Splitting Cut-Point Transitions

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

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