Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6, l7
Transitions:
t₀: l0(X₀, X₁, X₂) → l2(X₀, X₁, X₂)
t₂: l1(X₀, X₁, X₂) → l3(X₀, X₁, 0) :|: X₀ < 5
t₃: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: 5 ≤ X₀
t₁: l2(X₀, X₁, X₂) → l1(X₁, X₁, X₂)
t₅: l3(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₀ ≤ 2
t₆: l3(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: 9 < X₂
t₄: l3(X₀, X₁, X₂) → l6(X₀, X₁, X₂) :|: 2 < X₀ ∧ X₂ ≤ 9
t₉: l4(X₀, X₁, X₂) → l7(X₀, X₁, X₂)
t₈: l5(X₀, X₁, X₂) → l1(X₀+1, X₁, X₂)
t₇: l6(X₀, X₁, X₂) → l3(X₀, X₁, X₂+1)

Preprocessing

Found invariant X₂ ≤ 9 ∧ X₁+X₂ ≤ 13 ∧ X₂ ≤ 6+X₀ ∧ X₀+X₂ ≤ 13 ∧ 0 ≤ X₂ ∧ X₁ ≤ 4+X₂ ∧ 3 ≤ X₀+X₂ ∧ X₀ ≤ 4+X₂ ∧ X₁ ≤ 4 ∧ X₁ ≤ X₀ ∧ X₀+X₁ ≤ 8 ∧ X₀ ≤ 4 ∧ 3 ≤ X₀ for location l6

Found invariant X₁ ≤ X₀ ∧ 5 ≤ X₀ for location l7

Found invariant 0 ≤ X₂ ∧ X₁ ≤ 2+X₂ ∧ X₀ ≤ 2+X₂ ∧ X₁ ≤ 4 ∧ X₁ ≤ X₀ ∧ X₀+X₁ ≤ 8 ∧ X₀ ≤ 4 for location l5

Found invariant X₁ ≤ X₀ for location l1

Found invariant X₁ ≤ X₀ ∧ 5 ≤ X₀ for location l4

Found invariant 0 ≤ X₂ ∧ X₁ ≤ 4+X₂ ∧ X₀ ≤ 4+X₂ ∧ X₁ ≤ 4 ∧ X₁ ≤ X₀ ∧ X₀+X₁ ≤ 8 ∧ X₀ ≤ 4 for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6, l7
Transitions:
t₀: l0(X₀, X₁, X₂) → l2(X₀, X₁, X₂)
t₂: l1(X₀, X₁, X₂) → l3(X₀, X₁, 0) :|: X₀ < 5 ∧ X₁ ≤ X₀
t₃: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: 5 ≤ X₀ ∧ X₁ ≤ X₀
t₁: l2(X₀, X₁, X₂) → l1(X₁, X₁, X₂)
t₅: l3(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₀ ≤ 2 ∧ 0 ≤ X₂ ∧ X₁ ≤ 4+X₂ ∧ X₀ ≤ 4+X₂ ∧ X₁ ≤ 4 ∧ X₁ ≤ X₀ ∧ X₀+X₁ ≤ 8 ∧ X₀ ≤ 4
t₆: l3(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: 9 < X₂ ∧ 0 ≤ X₂ ∧ X₁ ≤ 4+X₂ ∧ X₀ ≤ 4+X₂ ∧ X₁ ≤ 4 ∧ X₁ ≤ X₀ ∧ X₀+X₁ ≤ 8 ∧ X₀ ≤ 4
t₄: l3(X₀, X₁, X₂) → l6(X₀, X₁, X₂) :|: 2 < X₀ ∧ X₂ ≤ 9 ∧ 0 ≤ X₂ ∧ X₁ ≤ 4+X₂ ∧ X₀ ≤ 4+X₂ ∧ X₁ ≤ 4 ∧ X₁ ≤ X₀ ∧ X₀+X₁ ≤ 8 ∧ X₀ ≤ 4
t₉: l4(X₀, X₁, X₂) → l7(X₀, X₁, X₂) :|: X₁ ≤ X₀ ∧ 5 ≤ X₀
t₈: l5(X₀, X₁, X₂) → l1(X₀+1, X₁, X₂) :|: 0 ≤ X₂ ∧ X₁ ≤ 2+X₂ ∧ X₀ ≤ 2+X₂ ∧ X₁ ≤ 4 ∧ X₁ ≤ X₀ ∧ X₀+X₁ ≤ 8 ∧ X₀ ≤ 4
t₇: l6(X₀, X₁, X₂) → l3(X₀, X₁, X₂+1) :|: X₂ ≤ 9 ∧ X₁+X₂ ≤ 13 ∧ X₂ ≤ 6+X₀ ∧ X₀+X₂ ≤ 13 ∧ 0 ≤ X₂ ∧ X₁ ≤ 4+X₂ ∧ 3 ≤ X₀+X₂ ∧ X₀ ≤ 4+X₂ ∧ X₁ ≤ 4 ∧ X₁ ≤ X₀ ∧ X₀+X₁ ≤ 8 ∧ X₀ ≤ 4 ∧ 3 ≤ X₀

MPRF for transition t₂: l1(X₀, X₁, X₂) → l3(X₀, X₁, 0) :|: X₀ < 5 ∧ X₁ ≤ X₀ of depth 1:

new bound:

X₁+5 {O(n)}

MPRF:

l5 [4-X₀ ]
l1 [5-X₀ ]
l6 [4-X₀ ]
l3 [4-X₀ ]

MPRF for transition t₅: l3(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₀ ≤ 2 ∧ 0 ≤ X₂ ∧ X₁ ≤ 4+X₂ ∧ X₀ ≤ 4+X₂ ∧ X₁ ≤ 4 ∧ X₁ ≤ X₀ ∧ X₀+X₁ ≤ 8 ∧ X₀ ≤ 4 of depth 1:

new bound:

X₁+5 {O(n)}

MPRF:

l5 [4-X₀ ]
l1 [5-X₀ ]
l6 [5-X₀ ]
l3 [5-X₀ ]

MPRF for transition t₆: l3(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: 9 < X₂ ∧ 0 ≤ X₂ ∧ X₁ ≤ 4+X₂ ∧ X₀ ≤ 4+X₂ ∧ X₁ ≤ 4 ∧ X₁ ≤ X₀ ∧ X₀+X₁ ≤ 8 ∧ X₀ ≤ 4 of depth 1:

new bound:

X₁+5 {O(n)}

MPRF:

l5 [4-X₀ ]
l1 [5-X₀ ]
l6 [5-X₀ ]
l3 [5-X₀ ]

MPRF for transition t₈: l5(X₀, X₁, X₂) → l1(X₀+1, X₁, X₂) :|: 0 ≤ X₂ ∧ X₁ ≤ 2+X₂ ∧ X₀ ≤ 2+X₂ ∧ X₁ ≤ 4 ∧ X₁ ≤ X₀ ∧ X₀+X₁ ≤ 8 ∧ X₀ ≤ 4 of depth 1:

new bound:

X₁+5 {O(n)}

MPRF:

l5 [5-X₀ ]
l1 [5-X₀ ]
l6 [5-X₀ ]
l3 [5-X₀ ]

Found invariant X₂ ≤ 9 ∧ X₁+X₂ ≤ 13 ∧ X₂ ≤ 6+X₀ ∧ X₀+X₂ ≤ 13 ∧ 0 ≤ X₂ ∧ X₁ ≤ 4+X₂ ∧ 3 ≤ X₀+X₂ ∧ X₀ ≤ 4+X₂ ∧ X₁ ≤ 4 ∧ X₁ ≤ X₀ ∧ X₀+X₁ ≤ 8 ∧ X₀ ≤ 4 ∧ 3 ≤ X₀ for location l6

Found invariant X₁ ≤ X₀ ∧ 5 ≤ X₀ for location l7

Found invariant 0 ≤ X₂ ∧ X₁ ≤ 2+X₂ ∧ X₀ ≤ 2+X₂ ∧ X₁ ≤ 4 ∧ X₁ ≤ X₀ ∧ X₀+X₁ ≤ 8 ∧ X₀ ≤ 4 for location l5

Found invariant X₁ ≤ X₀ for location l1

Found invariant X₁ ≤ X₀ ∧ 5 ≤ X₀ for location l4

Found invariant X₂ ≤ 10 ∧ X₁+X₂ ≤ 14 ∧ X₀+X₂ ≤ 14 ∧ 0 ≤ X₂ ∧ X₁ ≤ 4+X₂ ∧ X₀ ≤ 4+X₂ ∧ X₁ ≤ 4 ∧ X₁ ≤ X₀ ∧ X₀+X₁ ≤ 8 ∧ X₀ ≤ 4 for location l3

Time-Bound by TWN-Loops:

TWN-Loops: t₄ 23⋅X₁+115 {O(n)}

TWN-Loops:

entry: t₂: l1(X₀, X₁, X₂) → l3(X₀, X₁, 0) :|: X₀ < 5 ∧ X₁ ≤ X₀
results in twn-loop: twn:Inv: [0 ≤ X₂ ∧ X₁ ≤ 4+X₂ ∧ X₀ ≤ 4+X₂ ∧ X₁ ≤ 4 ∧ X₁ ≤ X₀ ∧ X₀+X₁ ≤ 8 ∧ X₀ ≤ 4 ∧ X₂ ≤ 9 ∧ X₁+X₂ ≤ 13 ∧ X₂ ≤ 6+X₀ ∧ X₀+X₂ ≤ 13 ∧ 0 ≤ X₂ ∧ X₁ ≤ 4+X₂ ∧ 3 ≤ X₀+X₂ ∧ X₀ ≤ 4+X₂ ∧ X₁ ≤ 4 ∧ X₁ ≤ X₀ ∧ X₀+X₁ ≤ 8 ∧ X₀ ≤ 4 ∧ 3 ≤ X₀] , (X₀,X₁,X₂) -> (X₀,X₁,X₂+1) :|: 2 < X₀ ∧ X₂ ≤ 9
order: [X₀; X₁; X₂]
closed-form:
X₀: X₀
X₁: X₁
X₂: X₂ + [[n != 0]] * n^1

Termination: true
Formula:

1 < 0 ∧ 2 < X₀
∨ X₂ < 9 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 2 < X₀
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₂ ≤ 9 ∧ 9 ≤ X₂ ∧ 2 < X₀

Stabilization-Threshold for: X₂ ≤ 9
alphas_abs: X₂+9
M: 0
N: 1
Bound: 2⋅X₂+20 {O(n)}

relevant size-bounds w.r.t. t₂:
X₂: 0 {O(1)}
Runtime-bound of t₂: X₁+5 {O(n)}
Results in: 23⋅X₁+115 {O(n)}

23⋅X₁+115 {O(n)}

Time-Bound by TWN-Loops:

TWN-Loops: t₇ 23⋅X₁+115 {O(n)}

relevant size-bounds w.r.t. t₂:
X₂: 0 {O(1)}
Runtime-bound of t₂: X₁+5 {O(n)}
Results in: 23⋅X₁+115 {O(n)}

23⋅X₁+115 {O(n)}

All Bounds

Timebounds

Overall timebound:50⋅X₁+254 {O(n)}
t₀: 1 {O(1)}
t₂: X₁+5 {O(n)}
t₃: 1 {O(1)}
t₁: 1 {O(1)}
t₄: 23⋅X₁+115 {O(n)}
t₅: X₁+5 {O(n)}
t₆: X₁+5 {O(n)}
t₉: 1 {O(1)}
t₈: X₁+5 {O(n)}
t₇: 23⋅X₁+115 {O(n)}

Costbounds

Overall costbound: 50⋅X₁+254 {O(n)}
t₀: 1 {O(1)}
t₂: X₁+5 {O(n)}
t₃: 1 {O(1)}
t₁: 1 {O(1)}
t₄: 23⋅X₁+115 {O(n)}
t₅: X₁+5 {O(n)}
t₆: X₁+5 {O(n)}
t₉: 1 {O(1)}
t₈: X₁+5 {O(n)}
t₇: 23⋅X₁+115 {O(n)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₂, X₀: 2⋅X₁+9 {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: 0 {O(1)}
t₃, X₀: 3⋅X₁+9 {O(n)}
t₃, X₁: 2⋅X₁ {O(n)}
t₃, X₂: X₂+10 {O(n)}
t₁, X₀: X₁ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₄, X₀: 4 {O(1)}
t₄, X₁: X₁ {O(n)}
t₄, X₂: 9 {O(1)}
t₅, X₀: 2⋅X₁+9 {O(n)}
t₅, X₁: X₁ {O(n)}
t₅, X₂: 0 {O(1)}
t₆, X₀: 4 {O(1)}
t₆, X₁: X₁ {O(n)}
t₆, X₂: 10 {O(1)}
t₉, X₀: 3⋅X₁+9 {O(n)}
t₉, X₁: 2⋅X₁ {O(n)}
t₉, X₂: X₂+10 {O(n)}
t₈, X₀: 2⋅X₁+9 {O(n)}
t₈, X₁: X₁ {O(n)}
t₈, X₂: 10 {O(1)}
t₇, X₀: 4 {O(1)}
t₇, X₁: X₁ {O(n)}
t₇, X₂: 10 {O(1)}