LTS Termination Proof

by AProVE

Input

Integer Transition System
• Initial Location: f207_0_mod_LE, f1_0_main_Load, f152_0_gcd_EQ, __init
• Transitions: (pre-variables and post-variables)  f1_0_main_Load 1 f152_0_gcd_EQ: x1 = _arg1 ∧ x2 = _arg2 ∧ x1 = _arg1P ∧ x2 = _arg2P ∧ 0 ≤ _arg1 − 1 ∧ −1 ≤ _arg1P − 1 ∧ −1 ≤ _arg2 − 1 ∧ −1 ≤ _arg2P − 1 f152_0_gcd_EQ 2 f207_0_mod_LE: x1 = _x ∧ x2 = _x1 ∧ x1 = _x2 ∧ x2 = _x3 ∧ _x1 = _x3 ∧ _x = _x2 ∧ 0 ≤ _x − 1 ∧ _x ≤ _x1 − 1 ∧ 0 ≤ _x1 − 1 f152_0_gcd_EQ 3 f207_0_mod_LE: x1 = _x4 ∧ x2 = _x5 ∧ x1 = _x6 ∧ x2 = _x7 ∧ _x5 = _x7 ∧ _x4 = _x6 ∧ 0 ≤ _x4 − 1 ∧ _x5 ≤ _x4 − 1 ∧ 0 ≤ _x5 − 1 f152_0_gcd_EQ 4 f152_0_gcd_EQ: x1 = _x8 ∧ x2 = _x9 ∧ x1 = _x10 ∧ x2 = _x11 ∧ 0 = _x11 ∧ _x9 = _x10 ∧ 0 = _x8 ∧ 0 ≤ _x9 − 1 f152_0_gcd_EQ 5 f152_0_gcd_EQ: x1 = _x12 ∧ x2 = _x13 ∧ x1 = _x14 ∧ x2 = _x15 ∧ 0 = _x15 ∧ _x12 = _x14 ∧ _x12 = _x13 ∧ 0 ≤ _x12 − 1 f207_0_mod_LE 6 f207_0_mod_LE: x1 = _x16 ∧ x2 = _x17 ∧ x1 = _x18 ∧ x2 = _x19 ∧ _x17 = _x19 ∧ _x16 − _x17 = _x18 ∧ 0 ≤ _x17 − 1 ∧ 0 ≤ _x16 − 1 ∧ _x17 ≤ _x16 − 1 f207_0_mod_LE 7 f152_0_gcd_EQ: x1 = _x20 ∧ x2 = _x21 ∧ x1 = _x22 ∧ x2 = _x23 ∧ _x20 = _x23 ∧ _x21 = _x22 ∧ _x20 ≤ _x21 − 1 __init 8 f1_0_main_Load: x1 = _x24 ∧ x2 = _x25 ∧ x1 = _x26 ∧ x2 = _x27 ∧ 0 ≤ 0

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 f207_0_mod_LE f207_0_mod_LE f207_0_mod_LE: x1 = x1 ∧ x2 = x2 f1_0_main_Load f1_0_main_Load f1_0_main_Load: x1 = x1 ∧ x2 = x2 f152_0_gcd_EQ f152_0_gcd_EQ f152_0_gcd_EQ: x1 = x1 ∧ x2 = x2 __init __init __init: x1 = x1 ∧ x2 = x2
and for every transition t, a duplicate t is considered.

2 SCC Decomposition

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

2.1 SCC Subproblem 1/1

Here we consider the SCC { f207_0_mod_LE, f152_0_gcd_EQ }.

2.1.1 Transition Removal

We remove transitions 4, 5 using the following ranking functions, which are bounded by 0.

 f152_0_gcd_EQ: −1 + x2 f207_0_mod_LE: −1 + x2

2.1.2 Transition Removal

We remove transition 2 using the following ranking functions, which are bounded by 0.

 f152_0_gcd_EQ: −2 + x2 f207_0_mod_LE: −3 + x2

2.1.3 Transition Removal

We remove transition 3 using the following ranking functions, which are bounded by 0.

 f207_0_mod_LE: −2 + x2 f152_0_gcd_EQ: −2 + x1

2.1.4 Transition Removal

We remove transition 7 using the following ranking functions, which are bounded by 0.

 f207_0_mod_LE: 0 f152_0_gcd_EQ: −1

2.1.5 Transition Removal

We remove transition 6 using the following ranking functions, which are bounded by 0.

 f207_0_mod_LE: x1

2.1.6 Trivial Cooperation Program

There are no more "sharp" transitions in the cooperation program. Hence the cooperation termination is proved.

Tool configuration

AProVE

• version: AProVE Commit ID: unknown
• strategy: Statistics for single proof: 100.00 % (8 real / 0 unknown / 0 assumptions / 8 total proof steps)