LTS Termination Proof

by T2Cert

Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

0: TRUE
1: TRUE
2: 2 − oldX3_post ≤ 0 ∧ 2 − oldX3_0 ≤ 0
3: TRUE
4: TRUE
5: TRUE
6: TRUE
7: TRUE
8: TRUE
9: TRUE
10: TRUE
11: TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
0 24 0: − x3_post + x3_post ≤ 0 ∧ x3_post − x3_post ≤ 0 ∧ − x3_0 + x3_0 ≤ 0 ∧ x3_0 − x3_0 ≤ 0 ∧ − x2_post + x2_post ≤ 0 ∧ x2_post − x2_post ≤ 0 ∧ − x2_0 + x2_0 ≤ 0 ∧ x2_0 − x2_0 ≤ 0 ∧ − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − x0_post + x0_post ≤ 0 ∧ x0_post − x0_post ≤ 0 ∧ − x0_0 + x0_0 ≤ 0 ∧ x0_0 − x0_0 ≤ 0 ∧ − oldX7_post + oldX7_post ≤ 0 ∧ oldX7_post − oldX7_post ≤ 0 ∧ − oldX7_0 + oldX7_0 ≤ 0 ∧ oldX7_0 − oldX7_0 ≤ 0 ∧ − oldX6_post + oldX6_post ≤ 0 ∧ oldX6_post − oldX6_post ≤ 0 ∧ − oldX6_0 + oldX6_0 ≤ 0 ∧ oldX6_0 − oldX6_0 ≤ 0 ∧ − oldX5_post + oldX5_post ≤ 0 ∧ oldX5_post − oldX5_post ≤ 0 ∧ − oldX5_0 + oldX5_0 ≤ 0 ∧ oldX5_0 − oldX5_0 ≤ 0 ∧ − oldX4_post + oldX4_post ≤ 0 ∧ oldX4_post − oldX4_post ≤ 0 ∧ − oldX4_0 + oldX4_0 ≤ 0 ∧ oldX4_0 − oldX4_0 ≤ 0 ∧ − oldX3_post + oldX3_post ≤ 0 ∧ oldX3_post − oldX3_post ≤ 0 ∧ − oldX3_0 + oldX3_0 ≤ 0 ∧ oldX3_0 − oldX3_0 ≤ 0 ∧ − oldX2_post + oldX2_post ≤ 0 ∧ oldX2_post − oldX2_post ≤ 0 ∧ − oldX2_0 + oldX2_0 ≤ 0 ∧ oldX2_0 − oldX2_0 ≤ 0 ∧ − oldX1_post + oldX1_post ≤ 0 ∧ oldX1_post − oldX1_post ≤ 0 ∧ − oldX1_0 + oldX1_0 ≤ 0 ∧ oldX1_0 − oldX1_0 ≤ 0 ∧ − oldX0_post + oldX0_post ≤ 0 ∧ oldX0_post − oldX0_post ≤ 0 ∧ − oldX0_0 + oldX0_0 ≤ 0 ∧ oldX0_0 − oldX0_0 ≤ 0
5 31 5: − x3_post + x3_post ≤ 0 ∧ x3_post − x3_post ≤ 0 ∧ − x3_0 + x3_0 ≤ 0 ∧ x3_0 − x3_0 ≤ 0 ∧ − x2_post + x2_post ≤ 0 ∧ x2_post − x2_post ≤ 0 ∧ − x2_0 + x2_0 ≤ 0 ∧ x2_0 − x2_0 ≤ 0 ∧ − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − x0_post + x0_post ≤ 0 ∧ x0_post − x0_post ≤ 0 ∧ − x0_0 + x0_0 ≤ 0 ∧ x0_0 − x0_0 ≤ 0 ∧ − oldX7_post + oldX7_post ≤ 0 ∧ oldX7_post − oldX7_post ≤ 0 ∧ − oldX7_0 + oldX7_0 ≤ 0 ∧ oldX7_0 − oldX7_0 ≤ 0 ∧ − oldX6_post + oldX6_post ≤ 0 ∧ oldX6_post − oldX6_post ≤ 0 ∧ − oldX6_0 + oldX6_0 ≤ 0 ∧ oldX6_0 − oldX6_0 ≤ 0 ∧ − oldX5_post + oldX5_post ≤ 0 ∧ oldX5_post − oldX5_post ≤ 0 ∧ − oldX5_0 + oldX5_0 ≤ 0 ∧ oldX5_0 − oldX5_0 ≤ 0 ∧ − oldX4_post + oldX4_post ≤ 0 ∧ oldX4_post − oldX4_post ≤ 0 ∧ − oldX4_0 + oldX4_0 ≤ 0 ∧ oldX4_0 − oldX4_0 ≤ 0 ∧ − oldX3_post + oldX3_post ≤ 0 ∧ oldX3_post − oldX3_post ≤ 0 ∧ − oldX3_0 + oldX3_0 ≤ 0 ∧ oldX3_0 − oldX3_0 ≤ 0 ∧ − oldX2_post + oldX2_post ≤ 0 ∧ oldX2_post − oldX2_post ≤ 0 ∧ − oldX2_0 + oldX2_0 ≤ 0 ∧ oldX2_0 − oldX2_0 ≤ 0 ∧ − oldX1_post + oldX1_post ≤ 0 ∧ oldX1_post − oldX1_post ≤ 0 ∧ − oldX1_0 + oldX1_0 ≤ 0 ∧ oldX1_0 − oldX1_0 ≤ 0 ∧ − oldX0_post + oldX0_post ≤ 0 ∧ oldX0_post − oldX0_post ≤ 0 ∧ − oldX0_0 + oldX0_0 ≤ 0 ∧ oldX0_0 − oldX0_0 ≤ 0
and for every transition t, a duplicate t is considered.

3 Transition Removal

We remove transitions 0, 3, 4, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23 using the following ranking functions, which are bounded by −27.

11: 0
10: 0
9: 0
6: 0
4: 0
5: 0
3: 0
0: 0
2: 0
7: 0
1: 0
8: 0
11: −11
10: −12
9: −13
6: −14
4: −15
5: −15
5_var_snapshot: −15
5*: −15
3: −18
0: −19
2: −19
0_var_snapshot: −19
0*: −19
7: −23
1: −24
8: −25

4 Location Addition

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

0* 27 0: − x3_post + x3_post ≤ 0 ∧ x3_post − x3_post ≤ 0 ∧ − x3_0 + x3_0 ≤ 0 ∧ x3_0 − x3_0 ≤ 0 ∧ − x2_post + x2_post ≤ 0 ∧ x2_post − x2_post ≤ 0 ∧ − x2_0 + x2_0 ≤ 0 ∧ x2_0 − x2_0 ≤ 0 ∧ − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − x0_post + x0_post ≤ 0 ∧ x0_post − x0_post ≤ 0 ∧ − x0_0 + x0_0 ≤ 0 ∧ x0_0 − x0_0 ≤ 0 ∧ − oldX7_post + oldX7_post ≤ 0 ∧ oldX7_post − oldX7_post ≤ 0 ∧ − oldX7_0 + oldX7_0 ≤ 0 ∧ oldX7_0 − oldX7_0 ≤ 0 ∧ − oldX6_post + oldX6_post ≤ 0 ∧ oldX6_post − oldX6_post ≤ 0 ∧ − oldX6_0 + oldX6_0 ≤ 0 ∧ oldX6_0 − oldX6_0 ≤ 0 ∧ − oldX5_post + oldX5_post ≤ 0 ∧ oldX5_post − oldX5_post ≤ 0 ∧ − oldX5_0 + oldX5_0 ≤ 0 ∧ oldX5_0 − oldX5_0 ≤ 0 ∧ − oldX4_post + oldX4_post ≤ 0 ∧ oldX4_post − oldX4_post ≤ 0 ∧ − oldX4_0 + oldX4_0 ≤ 0 ∧ oldX4_0 − oldX4_0 ≤ 0 ∧ − oldX3_post + oldX3_post ≤ 0 ∧ oldX3_post − oldX3_post ≤ 0 ∧ − oldX3_0 + oldX3_0 ≤ 0 ∧ oldX3_0 − oldX3_0 ≤ 0 ∧ − oldX2_post + oldX2_post ≤ 0 ∧ oldX2_post − oldX2_post ≤ 0 ∧ − oldX2_0 + oldX2_0 ≤ 0 ∧ oldX2_0 − oldX2_0 ≤ 0 ∧ − oldX1_post + oldX1_post ≤ 0 ∧ oldX1_post − oldX1_post ≤ 0 ∧ − oldX1_0 + oldX1_0 ≤ 0 ∧ oldX1_0 − oldX1_0 ≤ 0 ∧ − oldX0_post + oldX0_post ≤ 0 ∧ oldX0_post − oldX0_post ≤ 0 ∧ − oldX0_0 + oldX0_0 ≤ 0 ∧ oldX0_0 − oldX0_0 ≤ 0

5 Location Addition

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

0 25 0_var_snapshot: − x3_post + x3_post ≤ 0 ∧ x3_post − x3_post ≤ 0 ∧ − x3_0 + x3_0 ≤ 0 ∧ x3_0 − x3_0 ≤ 0 ∧ − x2_post + x2_post ≤ 0 ∧ x2_post − x2_post ≤ 0 ∧ − x2_0 + x2_0 ≤ 0 ∧ x2_0 − x2_0 ≤ 0 ∧ − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − x0_post + x0_post ≤ 0 ∧ x0_post − x0_post ≤ 0 ∧ − x0_0 + x0_0 ≤ 0 ∧ x0_0 − x0_0 ≤ 0 ∧ − oldX7_post + oldX7_post ≤ 0 ∧ oldX7_post − oldX7_post ≤ 0 ∧ − oldX7_0 + oldX7_0 ≤ 0 ∧ oldX7_0 − oldX7_0 ≤ 0 ∧ − oldX6_post + oldX6_post ≤ 0 ∧ oldX6_post − oldX6_post ≤ 0 ∧ − oldX6_0 + oldX6_0 ≤ 0 ∧ oldX6_0 − oldX6_0 ≤ 0 ∧ − oldX5_post + oldX5_post ≤ 0 ∧ oldX5_post − oldX5_post ≤ 0 ∧ − oldX5_0 + oldX5_0 ≤ 0 ∧ oldX5_0 − oldX5_0 ≤ 0 ∧ − oldX4_post + oldX4_post ≤ 0 ∧ oldX4_post − oldX4_post ≤ 0 ∧ − oldX4_0 + oldX4_0 ≤ 0 ∧ oldX4_0 − oldX4_0 ≤ 0 ∧ − oldX3_post + oldX3_post ≤ 0 ∧ oldX3_post − oldX3_post ≤ 0 ∧ − oldX3_0 + oldX3_0 ≤ 0 ∧ oldX3_0 − oldX3_0 ≤ 0 ∧ − oldX2_post + oldX2_post ≤ 0 ∧ oldX2_post − oldX2_post ≤ 0 ∧ − oldX2_0 + oldX2_0 ≤ 0 ∧ oldX2_0 − oldX2_0 ≤ 0 ∧ − oldX1_post + oldX1_post ≤ 0 ∧ oldX1_post − oldX1_post ≤ 0 ∧ − oldX1_0 + oldX1_0 ≤ 0 ∧ oldX1_0 − oldX1_0 ≤ 0 ∧ − oldX0_post + oldX0_post ≤ 0 ∧ oldX0_post − oldX0_post ≤ 0 ∧ − oldX0_0 + oldX0_0 ≤ 0 ∧ oldX0_0 − oldX0_0 ≤ 0

6 Location Addition

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

5* 34 5: − x3_post + x3_post ≤ 0 ∧ x3_post − x3_post ≤ 0 ∧ − x3_0 + x3_0 ≤ 0 ∧ x3_0 − x3_0 ≤ 0 ∧ − x2_post + x2_post ≤ 0 ∧ x2_post − x2_post ≤ 0 ∧ − x2_0 + x2_0 ≤ 0 ∧ x2_0 − x2_0 ≤ 0 ∧ − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − x0_post + x0_post ≤ 0 ∧ x0_post − x0_post ≤ 0 ∧ − x0_0 + x0_0 ≤ 0 ∧ x0_0 − x0_0 ≤ 0 ∧ − oldX7_post + oldX7_post ≤ 0 ∧ oldX7_post − oldX7_post ≤ 0 ∧ − oldX7_0 + oldX7_0 ≤ 0 ∧ oldX7_0 − oldX7_0 ≤ 0 ∧ − oldX6_post + oldX6_post ≤ 0 ∧ oldX6_post − oldX6_post ≤ 0 ∧ − oldX6_0 + oldX6_0 ≤ 0 ∧ oldX6_0 − oldX6_0 ≤ 0 ∧ − oldX5_post + oldX5_post ≤ 0 ∧ oldX5_post − oldX5_post ≤ 0 ∧ − oldX5_0 + oldX5_0 ≤ 0 ∧ oldX5_0 − oldX5_0 ≤ 0 ∧ − oldX4_post + oldX4_post ≤ 0 ∧ oldX4_post − oldX4_post ≤ 0 ∧ − oldX4_0 + oldX4_0 ≤ 0 ∧ oldX4_0 − oldX4_0 ≤ 0 ∧ − oldX3_post + oldX3_post ≤ 0 ∧ oldX3_post − oldX3_post ≤ 0 ∧ − oldX3_0 + oldX3_0 ≤ 0 ∧ oldX3_0 − oldX3_0 ≤ 0 ∧ − oldX2_post + oldX2_post ≤ 0 ∧ oldX2_post − oldX2_post ≤ 0 ∧ − oldX2_0 + oldX2_0 ≤ 0 ∧ oldX2_0 − oldX2_0 ≤ 0 ∧ − oldX1_post + oldX1_post ≤ 0 ∧ oldX1_post − oldX1_post ≤ 0 ∧ − oldX1_0 + oldX1_0 ≤ 0 ∧ oldX1_0 − oldX1_0 ≤ 0 ∧ − oldX0_post + oldX0_post ≤ 0 ∧ oldX0_post − oldX0_post ≤ 0 ∧ − oldX0_0 + oldX0_0 ≤ 0 ∧ oldX0_0 − oldX0_0 ≤ 0

7 Location Addition

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

5 32 5_var_snapshot: − x3_post + x3_post ≤ 0 ∧ x3_post − x3_post ≤ 0 ∧ − x3_0 + x3_0 ≤ 0 ∧ x3_0 − x3_0 ≤ 0 ∧ − x2_post + x2_post ≤ 0 ∧ x2_post − x2_post ≤ 0 ∧ − x2_0 + x2_0 ≤ 0 ∧ x2_0 − x2_0 ≤ 0 ∧ − x1_post + x1_post ≤ 0 ∧ x1_post − x1_post ≤ 0 ∧ − x1_0 + x1_0 ≤ 0 ∧ x1_0 − x1_0 ≤ 0 ∧ − x0_post + x0_post ≤ 0 ∧ x0_post − x0_post ≤ 0 ∧ − x0_0 + x0_0 ≤ 0 ∧ x0_0 − x0_0 ≤ 0 ∧ − oldX7_post + oldX7_post ≤ 0 ∧ oldX7_post − oldX7_post ≤ 0 ∧ − oldX7_0 + oldX7_0 ≤ 0 ∧ oldX7_0 − oldX7_0 ≤ 0 ∧ − oldX6_post + oldX6_post ≤ 0 ∧ oldX6_post − oldX6_post ≤ 0 ∧ − oldX6_0 + oldX6_0 ≤ 0 ∧ oldX6_0 − oldX6_0 ≤ 0 ∧ − oldX5_post + oldX5_post ≤ 0 ∧ oldX5_post − oldX5_post ≤ 0 ∧ − oldX5_0 + oldX5_0 ≤ 0 ∧ oldX5_0 − oldX5_0 ≤ 0 ∧ − oldX4_post + oldX4_post ≤ 0 ∧ oldX4_post − oldX4_post ≤ 0 ∧ − oldX4_0 + oldX4_0 ≤ 0 ∧ oldX4_0 − oldX4_0 ≤ 0 ∧ − oldX3_post + oldX3_post ≤ 0 ∧ oldX3_post − oldX3_post ≤ 0 ∧ − oldX3_0 + oldX3_0 ≤ 0 ∧ oldX3_0 − oldX3_0 ≤ 0 ∧ − oldX2_post + oldX2_post ≤ 0 ∧ oldX2_post − oldX2_post ≤ 0 ∧ − oldX2_0 + oldX2_0 ≤ 0 ∧ oldX2_0 − oldX2_0 ≤ 0 ∧ − oldX1_post + oldX1_post ≤ 0 ∧ oldX1_post − oldX1_post ≤ 0 ∧ − oldX1_0 + oldX1_0 ≤ 0 ∧ oldX1_0 − oldX1_0 ≤ 0 ∧ − oldX0_post + oldX0_post ≤ 0 ∧ oldX0_post − oldX0_post ≤ 0 ∧ − oldX0_0 + oldX0_0 ≤ 0 ∧ oldX0_0 − oldX0_0 ≤ 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 { 0, 2, 0_var_snapshot, 0* }.

8.1.1 Transition Removal

We remove transition 1 using the following ranking functions, which are bounded by 5.

0: −1 + 4⋅x3_0
2: 1 + 4⋅x3_0
0_var_snapshot: −2 + 4⋅x3_0
0*: 4⋅x3_0

8.1.2 Transition Removal

We remove transitions 27, 2 using the following ranking functions, which are bounded by −2.

0: −1
2: oldX3_0
0_var_snapshot: −2
0*: 0

8.1.3 Transition Removal

We remove transition 25 using the following ranking functions, which are bounded by −1.

0: 0
2: 0
0_var_snapshot: −1
0*: 0

8.1.4 Splitting Cut-Point Transitions

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

8.1.4.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 24.

8.1.4.1.1 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 { 4, 5, 5_var_snapshot, 5* }.

8.2.1 Transition Removal

We remove transition 7 using the following ranking functions, which are bounded by 23.

4: −8 + 3⋅x2_0
5: 3⋅x2_0
5_var_snapshot: 3⋅x2_0
5*: 8 + 3⋅x2_0

8.2.2 Transition Removal

We remove transitions 32, 5 using the following ranking functions, which are bounded by −1.

4: 2
5: 0
5_var_snapshot: −1
5*: 1

8.2.3 Transition Removal

We remove transition 34 using the following ranking functions, which are bounded by 0.

4: 0
5: 0
5_var_snapshot: 0
5*: 1

8.2.4 Splitting Cut-Point Transitions

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

8.2.4.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 31.

8.2.4.1.1 Splitting Cut-Point Transitions

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

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