Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄)
t₂: l1(X₀, X₁, X₂, X₃, X₄) → l3(X₀, X₁, X₂, X₃, X₄) :|: 0 < X₀ ∧ 0 < X₁
t₃: l1(X₀, X₁, X₂, X₃, X₄) → l4(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ 0
t₄: l1(X₀, X₁, X₂, X₃, X₄) → l4(X₀, X₁, X₂, X₃, X₄) :|: X₁ ≤ 0
t₁: l2(X₀, X₁, X₂, X₃, X₄) → l1(X₃, X₄, X₂, X₃, X₄)
t₅: l3(X₀, X₁, X₂, X₃, X₄) → l1(X₀-1, X₁-1, X₂, X₃, X₄)
t₆: l4(X₀, X₁, X₂, X₃, X₄) → l5(X₀, X₁, X₂, X₃, X₄)

Preprocessing

Eliminate variables {X₂} that do not contribute to the problem

Found invariant X₁ ≤ X₄ ∧ X₀ ≤ X₃ for location l5

Found invariant X₁ ≤ X₄ ∧ X₀ ≤ X₃ for location l1

Found invariant X₁ ≤ X₄ ∧ X₀ ≤ X₃ for location l4

Found invariant 1 ≤ X₄ ∧ 2 ≤ X₃+X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ 1 ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₃, X₄
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₁₄: l0(X₀, X₁, X₃, X₄) → l2(X₀, X₁, X₃, X₄)
t₁₅: l1(X₀, X₁, X₃, X₄) → l3(X₀, X₁, X₃, X₄) :|: 0 < X₀ ∧ 0 < X₁ ∧ X₁ ≤ X₄ ∧ X₀ ≤ X₃
t₁₆: l1(X₀, X₁, X₃, X₄) → l4(X₀, X₁, X₃, X₄) :|: X₀ ≤ 0 ∧ X₁ ≤ X₄ ∧ X₀ ≤ X₃
t₁₇: l1(X₀, X₁, X₃, X₄) → l4(X₀, X₁, X₃, X₄) :|: X₁ ≤ 0 ∧ X₁ ≤ X₄ ∧ X₀ ≤ X₃
t₁₈: l2(X₀, X₁, X₃, X₄) → l1(X₃, X₄, X₃, X₄)
t₁₉: l3(X₀, X₁, X₃, X₄) → l1(X₀-1, X₁-1, X₃, X₄) :|: 1 ≤ X₄ ∧ 2 ≤ X₃+X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ 1 ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀
t₂₀: l4(X₀, X₁, X₃, X₄) → l5(X₀, X₁, X₃, X₄) :|: X₁ ≤ X₄ ∧ X₀ ≤ X₃

Found invariant X₁ ≤ X₄ ∧ X₀ ≤ X₃ for location l5

Found invariant X₁ ≤ X₄ ∧ X₀ ≤ X₃ for location l1

Found invariant X₁ ≤ X₄ ∧ X₀ ≤ X₃ for location l4

Found invariant 1 ≤ X₄ ∧ 2 ≤ X₃+X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ 1 ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l3

Time-Bound by TWN-Loops:

TWN-Loops: t₁₅ 2⋅X₃+2⋅X₄+6 {O(n)}

TWN-Loops:

entry: t₁₈: l2(X₀, X₁, X₃, X₄) → l1(X₃, X₄, X₃, X₄)
results in twn-loop: twn:Inv: [X₁ ≤ X₄ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₄ ∧ 2 ≤ X₃+X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ 1 ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀] , (X₀,X₁,X₃,X₄) -> (X₀-1,X₁-1,X₃,X₄) :|: 0 < X₀ ∧ 0 < X₁
order: [X₀; X₁; X₃; X₄]
closed-form:
X₀: X₀ + [[n != 0]] * -1 * n^1
X₁: X₁ + [[n != 0]] * -1 * n^1
X₃: X₃
X₄: X₄

Termination: true
Formula:

1 < 0 ∧ 2 < 0
∨ 1 < 0 ∧ 2 < X₀+X₁ ∧ 2 ≤ 0 ∧ 0 ≤ 2
∨ 1 < 0 ∧ 2 ≤ 0 ∧ 0 ≤ 2 ∧ 2 ≤ X₀+X₁ ∧ X₀+X₁ ≤ 2
∨ 1 < 0 ∧ 0 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 2 < 0
∨ 1 < 0 ∧ 0 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 2 < X₀+X₁ ∧ 2 ≤ 0 ∧ 0 ≤ 2
∨ 1 < 0 ∧ 0 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 2 ≤ 0 ∧ 0 ≤ 2 ∧ 2 ≤ X₀+X₁ ∧ X₀+X₁ ≤ 2
∨ 0 < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 < 0 ∧ 2 < 0
∨ 0 < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 < 0 ∧ 2 < X₀+X₁ ∧ 2 ≤ 0 ∧ 0 ≤ 2
∨ 0 < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 < 0 ∧ 2 ≤ 0 ∧ 0 ≤ 2 ∧ 2 ≤ X₀+X₁ ∧ X₀+X₁ ≤ 2
∨ 0 < X₁ ∧ 0 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 2 < 0
∨ 0 < X₁ ∧ 0 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 2 < X₀+X₁ ∧ 2 ≤ 0 ∧ 0 ≤ 2
∨ 0 < X₁ ∧ 0 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 2 ≤ 0 ∧ 0 ≤ 2 ∧ 2 ≤ X₀+X₁ ∧ X₀+X₁ ≤ 2

Stabilization-Threshold for: 0 < X₁
alphas_abs: X₁
M: 0
N: 1
Bound: 2⋅X₁+2 {O(n)}
Stabilization-Threshold for: 0 < X₀
alphas_abs: X₀
M: 0
N: 1
Bound: 2⋅X₀+2 {O(n)}

relevant size-bounds w.r.t. t₁₈:
X₀: X₃ {O(n)}
X₁: X₄ {O(n)}
Runtime-bound of t₁₈: 1 {O(1)}
Results in: 2⋅X₃+2⋅X₄+6 {O(n)}

2⋅X₃+2⋅X₄+6 {O(n)}

Time-Bound by TWN-Loops:

TWN-Loops: t₁₉ 2⋅X₃+2⋅X₄+6 {O(n)}

relevant size-bounds w.r.t. t₁₈:
X₀: X₃ {O(n)}
X₁: X₄ {O(n)}
Runtime-bound of t₁₈: 1 {O(1)}
Results in: 2⋅X₃+2⋅X₄+6 {O(n)}

2⋅X₃+2⋅X₄+6 {O(n)}

All Bounds

Timebounds

Overall timebound:4⋅X₃+4⋅X₄+17 {O(n)}
t₁₄: 1 {O(1)}
t₁₅: 2⋅X₃+2⋅X₄+6 {O(n)}
t₁₆: 1 {O(1)}
t₁₇: 1 {O(1)}
t₁₈: 1 {O(1)}
t₁₉: 2⋅X₃+2⋅X₄+6 {O(n)}
t₂₀: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₃+4⋅X₄+17 {O(n)}
t₁₄: 1 {O(1)}
t₁₅: 2⋅X₃+2⋅X₄+6 {O(n)}
t₁₆: 1 {O(1)}
t₁₇: 1 {O(1)}
t₁₈: 1 {O(1)}
t₁₉: 2⋅X₃+2⋅X₄+6 {O(n)}
t₂₀: 1 {O(1)}

Sizebounds

t₁₄, X₀: X₀ {O(n)}
t₁₄, X₁: X₁ {O(n)}
t₁₄, X₃: X₃ {O(n)}
t₁₄, X₄: X₄ {O(n)}
t₁₅, X₀: X₃ {O(n)}
t₁₅, X₁: X₄ {O(n)}
t₁₅, X₃: X₃ {O(n)}
t₁₅, X₄: X₄ {O(n)}
t₁₆, X₀: 2⋅X₃ {O(n)}
t₁₆, X₁: 2⋅X₄ {O(n)}
t₁₆, X₃: 2⋅X₃ {O(n)}
t₁₆, X₄: 2⋅X₄ {O(n)}
t₁₇, X₀: 2⋅X₃ {O(n)}
t₁₇, X₁: 2⋅X₄ {O(n)}
t₁₇, X₃: 2⋅X₃ {O(n)}
t₁₇, X₄: 2⋅X₄ {O(n)}
t₁₈, X₀: X₃ {O(n)}
t₁₈, X₁: X₄ {O(n)}
t₁₈, X₃: X₃ {O(n)}
t₁₈, X₄: X₄ {O(n)}
t₁₉, X₀: X₃ {O(n)}
t₁₉, X₁: X₄ {O(n)}
t₁₉, X₃: X₃ {O(n)}
t₁₉, X₄: X₄ {O(n)}
t₂₀, X₀: 4⋅X₃ {O(n)}
t₂₀, X₁: 4⋅X₄ {O(n)}
t₂₀, X₃: 4⋅X₃ {O(n)}
t₂₀, X₄: 4⋅X₄ {O(n)}