LTS Termination Proof

by T2Cert

Input

Integer Transition System
• Initial Location: 6
• Transitions: (pre-variables and post-variables)  0 0 1: 0 ≤ 0 ∧ 0 ≤ 0 ∧ i_13_post ≤ 0 ∧ − i_13_post ≤ 0 ∧ − i_13_post ≤ 0 ∧ i_13_post ≤ 0 ∧ i_13_0 − i_13_post ≤ 0 ∧ − i_13_0 + i_13_post ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − rt_11_post + rt_11_post ≤ 0 ∧ rt_11_post − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_0 ≤ 0 ∧ rt_11_0 − rt_11_0 ≤ 0 ∧ − i_21_post + i_21_post ≤ 0 ∧ i_21_post − i_21_post ≤ 0 ∧ − i_21_0 + i_21_0 ≤ 0 ∧ i_21_0 − i_21_0 ≤ 0 ∧ − i_13_1 + i_13_1 ≤ 0 ∧ i_13_1 − i_13_1 ≤ 0 ∧ − a_20_post + a_20_post ≤ 0 ∧ a_20_post − a_20_post ≤ 0 ∧ − a_20_0 + a_20_0 ≤ 0 ∧ a_20_0 − a_20_0 ≤ 0 ∧ − ___const_10_0 + ___const_10_0 ≤ 0 ∧ ___const_10_0 − ___const_10_0 ≤ 0 1 3 4: 0 ≤ 0 ∧ 0 ≤ 0 ∧ ___const_10_0 − i_13_0 ≤ 0 ∧ rt_11_post − st_14_0 ≤ 0 ∧ − rt_11_post + st_14_0 ≤ 0 ∧ rt_11_0 − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_post ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − i_21_post + i_21_post ≤ 0 ∧ i_21_post − i_21_post ≤ 0 ∧ − i_21_0 + i_21_0 ≤ 0 ∧ i_21_0 − i_21_0 ≤ 0 ∧ − i_13_post + i_13_post ≤ 0 ∧ i_13_post − i_13_post ≤ 0 ∧ − i_13_1 + i_13_1 ≤ 0 ∧ i_13_1 − i_13_1 ≤ 0 ∧ − i_13_0 + i_13_0 ≤ 0 ∧ i_13_0 − i_13_0 ≤ 0 ∧ − a_20_post + a_20_post ≤ 0 ∧ a_20_post − a_20_post ≤ 0 ∧ − a_20_0 + a_20_0 ≤ 0 ∧ a_20_0 − a_20_0 ≤ 0 ∧ − ___const_10_0 + ___const_10_0 ≤ 0 ∧ ___const_10_0 − ___const_10_0 ≤ 0 1 4 5: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 − ___const_10_0 + i_13_0 ≤ 0 ∧ −1 − i_13_0 + i_13_post ≤ 0 ∧ 1 + i_13_0 − i_13_post ≤ 0 ∧ −1 + i_13_post − i_21_post ≤ 0 ∧ 1 − i_13_post + i_21_post ≤ 0 ∧ 1 − ___const_10_0 + i_21_post ≤ 0 ∧ i_13_0 − i_13_post ≤ 0 ∧ − i_13_0 + i_13_post ≤ 0 ∧ i_21_0 − i_21_post ≤ 0 ∧ − i_21_0 + i_21_post ≤ 0 ∧ − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − rt_11_post + rt_11_post ≤ 0 ∧ rt_11_post − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_0 ≤ 0 ∧ rt_11_0 − rt_11_0 ≤ 0 ∧ − i_13_1 + i_13_1 ≤ 0 ∧ i_13_1 − i_13_1 ≤ 0 ∧ − a_20_post + a_20_post ≤ 0 ∧ a_20_post − a_20_post ≤ 0 ∧ − a_20_0 + a_20_0 ≤ 0 ∧ a_20_0 − a_20_0 ≤ 0 ∧ − ___const_10_0 + ___const_10_0 ≤ 0 ∧ ___const_10_0 − ___const_10_0 ≤ 0 5 5 1: − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − rt_11_post + rt_11_post ≤ 0 ∧ rt_11_post − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_0 ≤ 0 ∧ rt_11_0 − rt_11_0 ≤ 0 ∧ − i_21_post + i_21_post ≤ 0 ∧ i_21_post − i_21_post ≤ 0 ∧ − i_21_0 + i_21_0 ≤ 0 ∧ i_21_0 − i_21_0 ≤ 0 ∧ − i_13_post + i_13_post ≤ 0 ∧ i_13_post − i_13_post ≤ 0 ∧ − i_13_1 + i_13_1 ≤ 0 ∧ i_13_1 − i_13_1 ≤ 0 ∧ − i_13_0 + i_13_0 ≤ 0 ∧ i_13_0 − i_13_0 ≤ 0 ∧ − a_20_post + a_20_post ≤ 0 ∧ a_20_post − a_20_post ≤ 0 ∧ − a_20_0 + a_20_0 ≤ 0 ∧ a_20_0 − a_20_0 ≤ 0 ∧ − ___const_10_0 + ___const_10_0 ≤ 0 ∧ ___const_10_0 − ___const_10_0 ≤ 0 6 6 0: − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − rt_11_post + rt_11_post ≤ 0 ∧ rt_11_post − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_0 ≤ 0 ∧ rt_11_0 − rt_11_0 ≤ 0 ∧ − i_21_post + i_21_post ≤ 0 ∧ i_21_post − i_21_post ≤ 0 ∧ − i_21_0 + i_21_0 ≤ 0 ∧ i_21_0 − i_21_0 ≤ 0 ∧ − i_13_post + i_13_post ≤ 0 ∧ i_13_post − i_13_post ≤ 0 ∧ − i_13_1 + i_13_1 ≤ 0 ∧ i_13_1 − i_13_1 ≤ 0 ∧ − i_13_0 + i_13_0 ≤ 0 ∧ i_13_0 − i_13_0 ≤ 0 ∧ − a_20_post + a_20_post ≤ 0 ∧ a_20_post − a_20_post ≤ 0 ∧ − a_20_0 + a_20_0 ≤ 0 ∧ a_20_0 − a_20_0 ≤ 0 ∧ − ___const_10_0 + ___const_10_0 ≤ 0 ∧ ___const_10_0 − ___const_10_0 ≤ 0

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 1 7 1: − st_14_0 + st_14_0 ≤ 0 ∧ st_14_0 − st_14_0 ≤ 0 ∧ − rt_11_post + rt_11_post ≤ 0 ∧ rt_11_post − rt_11_post ≤ 0 ∧ − rt_11_0 + rt_11_0 ≤ 0 ∧ rt_11_0 − rt_11_0 ≤ 0 ∧ − i_21_post + i_21_post ≤ 0 ∧ i_21_post − i_21_post ≤ 0 ∧ − i_21_0 + i_21_0 ≤ 0 ∧ i_21_0 − i_21_0 ≤ 0 ∧ − i_13_post + i_13_post ≤ 0 ∧ i_13_post − i_13_post ≤ 0 ∧ − i_13_1 + i_13_1 ≤ 0 ∧ i_13_1 − i_13_1 ≤ 0 ∧ − i_13_0 + i_13_0 ≤ 0 ∧ i_13_0 − i_13_0 ≤ 0 ∧ − a_20_post + a_20_post ≤ 0 ∧ a_20_post − a_20_post ≤ 0 ∧ − a_20_0 + a_20_0 ≤ 0 ∧ a_20_0 − a_20_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.

2 Transition Removal

We remove transitions 0, 3, 6 using the following ranking functions, which are bounded by −13.

 6: 0 0: 0 1: 0 5: 0 4: 0 6: −5 0: −6 1: −7 5: −7 1_var_snapshot: −7 1*: −7 4: −11

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

1* 10 1: st_14_0 + st_14_0 ≤ 0st_14_0st_14_0 ≤ 0rt_11_post + rt_11_post ≤ 0rt_11_postrt_11_post ≤ 0rt_11_0 + rt_11_0 ≤ 0rt_11_0rt_11_0 ≤ 0i_21_post + i_21_post ≤ 0i_21_posti_21_post ≤ 0i_21_0 + i_21_0 ≤ 0i_21_0i_21_0 ≤ 0i_13_post + i_13_post ≤ 0i_13_posti_13_post ≤ 0i_13_1 + i_13_1 ≤ 0i_13_1i_13_1 ≤ 0i_13_0 + i_13_0 ≤ 0i_13_0i_13_0 ≤ 0a_20_post + a_20_post ≤ 0a_20_posta_20_post ≤ 0a_20_0 + a_20_0 ≤ 0a_20_0a_20_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 8 1_var_snapshot: st_14_0 + st_14_0 ≤ 0st_14_0st_14_0 ≤ 0rt_11_post + rt_11_post ≤ 0rt_11_postrt_11_post ≤ 0rt_11_0 + rt_11_0 ≤ 0rt_11_0rt_11_0 ≤ 0i_21_post + i_21_post ≤ 0i_21_posti_21_post ≤ 0i_21_0 + i_21_0 ≤ 0i_21_0i_21_0 ≤ 0i_13_post + i_13_post ≤ 0i_13_posti_13_post ≤ 0i_13_1 + i_13_1 ≤ 0i_13_1i_13_1 ≤ 0i_13_0 + i_13_0 ≤ 0i_13_0i_13_0 ≤ 0a_20_post + a_20_post ≤ 0a_20_posta_20_post ≤ 0a_20_0 + a_20_0 ≤ 0a_20_0a_20_0 ≤ 0___const_10_0 + ___const_10_0 ≤ 0___const_10_0___const_10_0 ≤ 0

5 SCC Decomposition

We consider subproblems for each of the 1 SCC(s) of the program graph.

5.1 SCC Subproblem 1/1

Here we consider the SCC { 1, 5, 1_var_snapshot, 1* }.

5.1.1 Transition Removal

We remove transition 4 using the following ranking functions, which are bounded by 0.

 1: −2 + 3⋅___const_10_0 − 3⋅i_13_0 5: 3⋅___const_10_0 − 3⋅i_13_0 1_var_snapshot: −2 + 3⋅___const_10_0 − 3⋅i_13_0 1*: −1 + 3⋅___const_10_0 − 3⋅i_13_0

5.1.2 Transition Removal

We remove transitions 8, 10, 5 using the following ranking functions, which are bounded by −1.

 1: 0 5: 2 1_var_snapshot: −1 1*: 1

5.1.3 Splitting Cut-Point Transitions

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

5.1.3.1 Cut-Point Subproblem 1/1

Here we consider cut-point transition 7.

5.1.3.1.1 Splitting Cut-Point Transitions

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

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