# 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 ∧ 1 − i4_0 + n3_0 ≤ 0 ∧ − Fnew5_0 + ans8_post ≤ 0 ∧ Fnew5_0 − ans8_post ≤ 0 ∧ − ans8_post + ret_fib9_post ≤ 0 ∧ ans8_post − ret_fib9_post ≤ 0 ∧ − ret_fib9_post + tmp_post ≤ 0 ∧ ret_fib9_post − tmp_post ≤ 0 ∧ ans8_0 − ans8_post ≤ 0 ∧ − ans8_0 + ans8_post ≤ 0 ∧ ret_fib9_0 − ret_fib9_post ≤ 0 ∧ − ret_fib9_0 + ret_fib9_post ≤ 0 ∧ tmp_0 − tmp_post ≤ 0 ∧ − tmp_0 + tmp_post ≤ 0 ∧ − temp7_post + temp7_post ≤ 0 ∧ temp7_post − temp7_post ≤ 0 ∧ − temp7_0 + temp7_0 ≤ 0 ∧ temp7_0 − temp7_0 ≤ 0 ∧ − n3_post + n3_post ≤ 0 ∧ n3_post − n3_post ≤ 0 ∧ − n3_0 + n3_0 ≤ 0 ∧ n3_0 − n3_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0 ∧ − a_post + a_post ≤ 0 ∧ a_post − a_post ≤ 0 ∧ − a_0 + a_0 ≤ 0 ∧ a_0 − a_0 ≤ 0 ∧ − Fold6_post + Fold6_post ≤ 0 ∧ Fold6_post − Fold6_post ≤ 0 ∧ − Fold6_0 + Fold6_0 ≤ 0 ∧ Fold6_0 − Fold6_0 ≤ 0 ∧ − Fnew5_post + Fnew5_post ≤ 0 ∧ Fnew5_post − Fnew5_post ≤ 0 ∧ − Fnew5_0 + Fnew5_0 ≤ 0 ∧ Fnew5_0 − Fnew5_0 ≤ 0 0 1 2: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ i4_0 − n3_0 ≤ 0 ∧ − Fnew5_0 + temp7_post ≤ 0 ∧ Fnew5_0 − temp7_post ≤ 0 ∧ − Fnew5_0 + Fnew5_post − Fold6_0 ≤ 0 ∧ Fnew5_0 − Fnew5_post + Fold6_0 ≤ 0 ∧ Fold6_post − temp7_post ≤ 0 ∧ − Fold6_post + temp7_post ≤ 0 ∧ −1 − i4_0 + i4_post ≤ 0 ∧ 1 + i4_0 − i4_post ≤ 0 ∧ Fnew5_0 − Fnew5_post ≤ 0 ∧ − Fnew5_0 + Fnew5_post ≤ 0 ∧ Fold6_0 − Fold6_post ≤ 0 ∧ − Fold6_0 + Fold6_post ≤ 0 ∧ i4_0 − i4_post ≤ 0 ∧ − i4_0 + i4_post ≤ 0 ∧ temp7_0 − temp7_post ≤ 0 ∧ − temp7_0 + temp7_post ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − ret_fib9_post + ret_fib9_post ≤ 0 ∧ ret_fib9_post − ret_fib9_post ≤ 0 ∧ − ret_fib9_0 + ret_fib9_0 ≤ 0 ∧ ret_fib9_0 − ret_fib9_0 ≤ 0 ∧ − n3_post + n3_post ≤ 0 ∧ n3_post − n3_post ≤ 0 ∧ − n3_0 + n3_0 ≤ 0 ∧ n3_0 − n3_0 ≤ 0 ∧ − ans8_post + ans8_post ≤ 0 ∧ ans8_post − ans8_post ≤ 0 ∧ − ans8_0 + ans8_0 ≤ 0 ∧ ans8_0 − ans8_0 ≤ 0 ∧ − a_post + a_post ≤ 0 ∧ a_post − a_post ≤ 0 ∧ − a_0 + a_0 ≤ 0 ∧ a_0 − a_0 ≤ 0 2 2 0: − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − temp7_post + temp7_post ≤ 0 ∧ temp7_post − temp7_post ≤ 0 ∧ − temp7_0 + temp7_0 ≤ 0 ∧ temp7_0 − temp7_0 ≤ 0 ∧ − ret_fib9_post + ret_fib9_post ≤ 0 ∧ ret_fib9_post − ret_fib9_post ≤ 0 ∧ − ret_fib9_0 + ret_fib9_0 ≤ 0 ∧ ret_fib9_0 − ret_fib9_0 ≤ 0 ∧ − n3_post + n3_post ≤ 0 ∧ n3_post − n3_post ≤ 0 ∧ − n3_0 + n3_0 ≤ 0 ∧ n3_0 − n3_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0 ∧ − ans8_post + ans8_post ≤ 0 ∧ ans8_post − ans8_post ≤ 0 ∧ − ans8_0 + ans8_0 ≤ 0 ∧ ans8_0 − ans8_0 ≤ 0 ∧ − a_post + a_post ≤ 0 ∧ a_post − a_post ≤ 0 ∧ − a_0 + a_0 ≤ 0 ∧ a_0 − a_0 ≤ 0 ∧ − Fold6_post + Fold6_post ≤ 0 ∧ Fold6_post − Fold6_post ≤ 0 ∧ − Fold6_0 + Fold6_0 ≤ 0 ∧ Fold6_0 − Fold6_0 ≤ 0 ∧ − Fnew5_post + Fnew5_post ≤ 0 ∧ Fnew5_post − Fnew5_post ≤ 0 ∧ − Fnew5_0 + Fnew5_0 ≤ 0 ∧ Fnew5_0 − Fnew5_0 ≤ 0 3 3 2: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ −30 + a_post ≤ 0 ∧ 30 − a_post ≤ 0 ∧ − a_post + n3_post ≤ 0 ∧ a_post − n3_post ≤ 0 ∧ −1 + Fnew5_post ≤ 0 ∧ 1 − Fnew5_post ≤ 0 ∧ Fold6_post ≤ 0 ∧ − Fold6_post ≤ 0 ∧ −2 + i4_post ≤ 0 ∧ 2 − i4_post ≤ 0 ∧ Fnew5_0 − Fnew5_post ≤ 0 ∧ − Fnew5_0 + Fnew5_post ≤ 0 ∧ Fold6_0 − Fold6_post ≤ 0 ∧ − Fold6_0 + Fold6_post ≤ 0 ∧ a_0 − a_post ≤ 0 ∧ − a_0 + a_post ≤ 0 ∧ i4_0 − i4_post ≤ 0 ∧ − i4_0 + i4_post ≤ 0 ∧ n3_0 − n3_post ≤ 0 ∧ − n3_0 + n3_post ≤ 0 ∧ − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − temp7_post + temp7_post ≤ 0 ∧ temp7_post − temp7_post ≤ 0 ∧ − temp7_0 + temp7_0 ≤ 0 ∧ temp7_0 − temp7_0 ≤ 0 ∧ − ret_fib9_post + ret_fib9_post ≤ 0 ∧ ret_fib9_post − ret_fib9_post ≤ 0 ∧ − ret_fib9_0 + ret_fib9_0 ≤ 0 ∧ ret_fib9_0 − ret_fib9_0 ≤ 0 ∧ − ans8_post + ans8_post ≤ 0 ∧ ans8_post − ans8_post ≤ 0 ∧ − ans8_0 + ans8_0 ≤ 0 ∧ ans8_0 − ans8_0 ≤ 0 4 4 3: − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − temp7_post + temp7_post ≤ 0 ∧ temp7_post − temp7_post ≤ 0 ∧ − temp7_0 + temp7_0 ≤ 0 ∧ temp7_0 − temp7_0 ≤ 0 ∧ − ret_fib9_post + ret_fib9_post ≤ 0 ∧ ret_fib9_post − ret_fib9_post ≤ 0 ∧ − ret_fib9_0 + ret_fib9_0 ≤ 0 ∧ ret_fib9_0 − ret_fib9_0 ≤ 0 ∧ − n3_post + n3_post ≤ 0 ∧ n3_post − n3_post ≤ 0 ∧ − n3_0 + n3_0 ≤ 0 ∧ n3_0 − n3_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0 ∧ − ans8_post + ans8_post ≤ 0 ∧ ans8_post − ans8_post ≤ 0 ∧ − ans8_0 + ans8_0 ≤ 0 ∧ ans8_0 − ans8_0 ≤ 0 ∧ − a_post + a_post ≤ 0 ∧ a_post − a_post ≤ 0 ∧ − a_0 + a_0 ≤ 0 ∧ a_0 − a_0 ≤ 0 ∧ − Fold6_post + Fold6_post ≤ 0 ∧ Fold6_post − Fold6_post ≤ 0 ∧ − Fold6_0 + Fold6_0 ≤ 0 ∧ Fold6_0 − Fold6_0 ≤ 0 ∧ − Fnew5_post + Fnew5_post ≤ 0 ∧ Fnew5_post − Fnew5_post ≤ 0 ∧ − Fnew5_0 + Fnew5_0 ≤ 0 ∧ Fnew5_0 − Fnew5_0 ≤ 0

## Proof

### 1 Invariant Updates

The following invariants are asserted.

 0: −30 + a_post ≤ 0 ∧ 30 − a_post ≤ 0 ∧ −30 + n3_post ≤ 0 ∧ −30 + a_0 ≤ 0 ∧ 30 − a_0 ≤ 0 ∧ −30 + n3_0 ≤ 0 1: −30 + a_post ≤ 0 ∧ 30 − a_post ≤ 0 ∧ −30 + n3_post ≤ 0 ∧ −30 + a_0 ≤ 0 ∧ 30 − a_0 ≤ 0 2: −30 + a_post ≤ 0 ∧ 30 − a_post ≤ 0 ∧ −30 + n3_post ≤ 0 ∧ −30 + a_0 ≤ 0 ∧ 30 − a_0 ≤ 0 ∧ −30 + n3_0 ≤ 0 3: TRUE 4: TRUE

The invariants are proved as follows.

### IMPACT Invariant Proof

• nodes (location) invariant:  0 (0) −30 + a_post ≤ 0 ∧ 30 − a_post ≤ 0 ∧ −30 + n3_post ≤ 0 ∧ −30 + a_0 ≤ 0 ∧ 30 − a_0 ≤ 0 ∧ −30 + n3_0 ≤ 0 1 (1) −30 + a_post ≤ 0 ∧ 30 − a_post ≤ 0 ∧ −30 + n3_post ≤ 0 ∧ −30 + a_0 ≤ 0 ∧ 30 − a_0 ≤ 0 2 (2) −30 + a_post ≤ 0 ∧ 30 − a_post ≤ 0 ∧ −30 + n3_post ≤ 0 ∧ −30 + a_0 ≤ 0 ∧ 30 − a_0 ≤ 0 ∧ −30 + n3_0 ≤ 0 3 (3) TRUE 4 (4) TRUE
• initial node: 4
• cover edges:
• transition edges:  0 0 1 0 1 2 2 2 0 3 3 2 4 4 3

### 2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 2 5 2: − tmp_post + tmp_post ≤ 0 ∧ tmp_post − tmp_post ≤ 0 ∧ − tmp_0 + tmp_0 ≤ 0 ∧ tmp_0 − tmp_0 ≤ 0 ∧ − temp7_post + temp7_post ≤ 0 ∧ temp7_post − temp7_post ≤ 0 ∧ − temp7_0 + temp7_0 ≤ 0 ∧ temp7_0 − temp7_0 ≤ 0 ∧ − ret_fib9_post + ret_fib9_post ≤ 0 ∧ ret_fib9_post − ret_fib9_post ≤ 0 ∧ − ret_fib9_0 + ret_fib9_0 ≤ 0 ∧ ret_fib9_0 − ret_fib9_0 ≤ 0 ∧ − n3_post + n3_post ≤ 0 ∧ n3_post − n3_post ≤ 0 ∧ − n3_0 + n3_0 ≤ 0 ∧ n3_0 − n3_0 ≤ 0 ∧ − i4_post + i4_post ≤ 0 ∧ i4_post − i4_post ≤ 0 ∧ − i4_0 + i4_0 ≤ 0 ∧ i4_0 − i4_0 ≤ 0 ∧ − ans8_post + ans8_post ≤ 0 ∧ ans8_post − ans8_post ≤ 0 ∧ − ans8_0 + ans8_0 ≤ 0 ∧ ans8_0 − ans8_0 ≤ 0 ∧ − a_post + a_post ≤ 0 ∧ a_post − a_post ≤ 0 ∧ − a_0 + a_0 ≤ 0 ∧ a_0 − a_0 ≤ 0 ∧ − Fold6_post + Fold6_post ≤ 0 ∧ Fold6_post − Fold6_post ≤ 0 ∧ − Fold6_0 + Fold6_0 ≤ 0 ∧ Fold6_0 − Fold6_0 ≤ 0 ∧ − Fnew5_post + Fnew5_post ≤ 0 ∧ Fnew5_post − Fnew5_post ≤ 0 ∧ − Fnew5_0 + Fnew5_0 ≤ 0 ∧ Fnew5_0 − Fnew5_0 ≤ 0
and for every transition t, a duplicate t is considered.

### 3 Transition Removal

We remove transitions 0, 3, 4 using the following ranking functions, which are bounded by −13.

 4: 0 3: 0 0: 0 2: 0 1: 0 4: −5 3: −6 0: −7 2: −7 2_var_snapshot: −7 2*: −7 1: −8
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, 0, 0, 0, 0, 0, 0, 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, 0, 0, 0, 0, 0, 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 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, 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, 0, 0, 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, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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.

2* 8 2: tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0temp7_post + temp7_post ≤ 0temp7_posttemp7_post ≤ 0temp7_0 + temp7_0 ≤ 0temp7_0temp7_0 ≤ 0ret_fib9_post + ret_fib9_post ≤ 0ret_fib9_postret_fib9_post ≤ 0ret_fib9_0 + ret_fib9_0 ≤ 0ret_fib9_0ret_fib9_0 ≤ 0n3_post + n3_post ≤ 0n3_postn3_post ≤ 0n3_0 + n3_0 ≤ 0n3_0n3_0 ≤ 0i4_post + i4_post ≤ 0i4_posti4_post ≤ 0i4_0 + i4_0 ≤ 0i4_0i4_0 ≤ 0ans8_post + ans8_post ≤ 0ans8_postans8_post ≤ 0ans8_0 + ans8_0 ≤ 0ans8_0ans8_0 ≤ 0a_post + a_post ≤ 0a_posta_post ≤ 0a_0 + a_0 ≤ 0a_0a_0 ≤ 0Fold6_post + Fold6_post ≤ 0Fold6_postFold6_post ≤ 0Fold6_0 + Fold6_0 ≤ 0Fold6_0Fold6_0 ≤ 0Fnew5_post + Fnew5_post ≤ 0Fnew5_postFnew5_post ≤ 0Fnew5_0 + Fnew5_0 ≤ 0Fnew5_0Fnew5_0 ≤ 0

### 5 Location Addition

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

2 6 2_var_snapshot: tmp_post + tmp_post ≤ 0tmp_posttmp_post ≤ 0tmp_0 + tmp_0 ≤ 0tmp_0tmp_0 ≤ 0temp7_post + temp7_post ≤ 0temp7_posttemp7_post ≤ 0temp7_0 + temp7_0 ≤ 0temp7_0temp7_0 ≤ 0ret_fib9_post + ret_fib9_post ≤ 0ret_fib9_postret_fib9_post ≤ 0ret_fib9_0 + ret_fib9_0 ≤ 0ret_fib9_0ret_fib9_0 ≤ 0n3_post + n3_post ≤ 0n3_postn3_post ≤ 0n3_0 + n3_0 ≤ 0n3_0n3_0 ≤ 0i4_post + i4_post ≤ 0i4_posti4_post ≤ 0i4_0 + i4_0 ≤ 0i4_0i4_0 ≤ 0ans8_post + ans8_post ≤ 0ans8_postans8_post ≤ 0ans8_0 + ans8_0 ≤ 0ans8_0ans8_0 ≤ 0a_post + a_post ≤ 0a_posta_post ≤ 0a_0 + a_0 ≤ 0a_0a_0 ≤ 0Fold6_post + Fold6_post ≤ 0Fold6_postFold6_post ≤ 0Fold6_0 + Fold6_0 ≤ 0Fold6_0Fold6_0 ≤ 0Fnew5_post + Fnew5_post ≤ 0Fnew5_postFnew5_post ≤ 0Fnew5_0 + Fnew5_0 ≤ 0Fnew5_0Fnew5_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, 2, 2_var_snapshot, 2* }.

### 6.1.1 Transition Removal

We remove transition 1 using the following ranking functions, which are bounded by −1021.

 0: − a_0 − 33⋅i4_0 2: 1 − 33⋅i4_0 2_var_snapshot: −33⋅i4_0 2*: 2 − 33⋅i4_0
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, 0, 0, 0, 33, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 8 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, 33, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 1 lexStrict[ [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 33, 0, 0, 0, 0, 0, 33, 0, 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, 33, 0, 0, 0, 0, 0, 0, 0, 0, 33, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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 lexWeak[ [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, 33, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0] ]

### 6.1.2 Transition Removal

We remove transitions 6, 2 using the following ranking functions, which are bounded by −31.

 0: − a_0 − a_post 2: 0 2_var_snapshot: − a_post 2*: a_post
Hints:
 6 lexStrict[ [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, 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, 0, 0, 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 lexWeak[ [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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 2 lexStrict[ [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, 1, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

### 6.1.3 Transition Removal

We remove transition 8 using the following ranking functions, which are bounded by 29.

 0: 0 2: 0 2_var_snapshot: 0 2*: a_0
Hints:
 8 lexStrict[ [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, 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, 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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