LTS Termination Proof

by T2Cert

Input

Integer Transition System

Proof

1 Invariant Updates

The following invariants are asserted.

0: −3000 + y_post ≤ 03000 − y_post ≤ 0−3000 + y_0 ≤ 03000 − y_0 ≤ 0
1: −3000 + y_post ≤ 03000 − y_post ≤ 03000 − y_0 ≤ 0−1999 + y_0 ≤ 0111 − x_post ≤ 0111 − x_0 ≤ 0
2: TRUE
3: TRUE

The invariants are proved as follows.

IMPACT Invariant Proof

2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
0 4 0: y_post + y_post ≤ 0y_posty_post ≤ 0y_0 + y_0 ≤ 0y_0y_0 ≤ 0x_post + x_post ≤ 0x_postx_post ≤ 0x_0 + x_0 ≤ 0x_0x_0 ≤ 0
and for every transition t, a duplicate t is considered.

3 Transition Removal

We remove transitions 2, 3 using the following ranking functions, which are bounded by −11.

3: 0
2: 0
0: 0
1: 0
3: −4
2: −5
0: −6
1: −6
0_var_snapshot: −6
0*: −6

4 Location Addition

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

0* 7 0: y_post + y_post ≤ 0y_posty_post ≤ 0y_0 + y_0 ≤ 0y_0y_0 ≤ 0x_post + x_post ≤ 0x_postx_post ≤ 0x_0 + x_0 ≤ 0x_0x_0 ≤ 0

5 Location Addition

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

0 5 0_var_snapshot: y_post + y_post ≤ 0y_posty_post ≤ 0y_0 + y_0 ≤ 0y_0y_0 ≤ 0x_post + x_post ≤ 0x_postx_post ≤ 0x_0 + x_0 ≤ 0x_0x_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, 1, 0_var_snapshot, 0* }.

6.1.1 Transition Removal

We remove transitions 7, 0, 1 using the following ranking functions, which are bounded by −15003001.

0: −15003000
1: 1001 − 4000⋅y_post
0_var_snapshot: −6002⋅y_0
0*: −4000⋅y_post

6.1.2 Transition Removal

We remove transition 5 using the following ranking functions, which are bounded by 0.

0: 1
1: 0
0_var_snapshot: 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 4.

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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