Initial Problem

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

Preprocessing

Found invariant X₃ ≤ X₀ for location l5

Found invariant X₃ ≤ X₀ for location l1

Found invariant X₃ ≤ X₀ for location l4

Found invariant X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ 1 ≤ X₂ ∧ X₀ ≤ X₁ for location l3

Problem after Preprocessing

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

Found invariant X₃ ≤ X₀ for location l5

Found invariant X₃ ≤ X₀ for location l1

Found invariant X₃ ≤ X₀ for location l4

Found invariant X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ 1 ≤ X₂ ∧ X₀ ≤ X₁ for location l3

Time-Bound by TWN-Loops:

TWN-Loops: t₂ 2⋅X₁+2⋅X₃+5 {O(n)}

TWN-Loops:

entry: t₁: l2(X₀, X₁, X₂, X₃) → l1(X₃, X₁, X₂, X₃)
results in twn-loop: twn:Inv: [X₃ ≤ X₀ ∧ X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ 1 ≤ X₂ ∧ X₀ ≤ X₁] , (X₀,X₁,X₂,X₃) -> (X₀+X₂,X₁,X₂,X₃) :|: 0 < X₂ ∧ X₀ ≤ X₁
order: [X₂; X₀; X₁; X₃]
closed-form:
X₂: X₂
X₀: X₀ + [[n != 0]] * X₂ * n^1
X₁: X₁
X₃: X₃

Termination: true
Formula:

X₂ < 0 ∧ 0 < X₂
∨ X₂ < 0 ∧ 0 < X₂ ∧ X₃ < X₀ ∧ 0 ≤ X₂ ∧ X₂ ≤ 0
∨ X₂ < 0 ∧ 0 < X₂ ∧ 0 ≤ X₂ ∧ X₂ ≤ 0 ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃
∨ X₀ < X₁ ∧ X₂ ≤ 0 ∧ 0 ≤ X₂ ∧ 0 < X₂
∨ X₀ < X₁ ∧ 0 < X₂ ∧ X₃ < X₀ ∧ 0 ≤ X₂ ∧ X₂ ≤ 0
∨ X₀ < X₁ ∧ 0 < X₂ ∧ 0 ≤ X₂ ∧ X₂ ≤ 0 ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃
∨ X₂ ≤ 0 ∧ 0 ≤ X₂ ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 0 < X₂
∨ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 0 < X₂ ∧ X₃ < X₀ ∧ 0 ≤ X₂ ∧ X₂ ≤ 0
∨ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 0 < X₂ ∧ 0 ≤ X₂ ∧ X₂ ≤ 0 ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃

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

relevant size-bounds w.r.t. t₁:
X₀: X₃ {O(n)}
X₁: X₁ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 2⋅X₁+2⋅X₃+5 {O(n)}

2⋅X₁+2⋅X₃+5 {O(n)}

Time-Bound by TWN-Loops:

TWN-Loops: t₅ 2⋅X₁+2⋅X₃+5 {O(n)}

relevant size-bounds w.r.t. t₁:
X₀: X₃ {O(n)}
X₁: X₁ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 2⋅X₁+2⋅X₃+5 {O(n)}

2⋅X₁+2⋅X₃+5 {O(n)}

All Bounds

Timebounds

Overall timebound:4⋅X₁+4⋅X₃+15 {O(n)}
t₀: 1 {O(1)}
t₂: 2⋅X₁+2⋅X₃+5 {O(n)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₁: 1 {O(1)}
t₅: 2⋅X₁+2⋅X₃+5 {O(n)}
t₆: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₁+4⋅X₃+15 {O(n)}
t₀: 1 {O(1)}
t₂: 2⋅X₁+2⋅X₃+5 {O(n)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₁: 1 {O(1)}
t₅: 2⋅X₁+2⋅X₃+5 {O(n)}
t₆: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₂, X₀: 2⋅X₁⋅X₂+2⋅X₂⋅X₃+6⋅X₂+X₃ {O(n^2)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₃, X₀: X₃ {O(n)}
t₃, X₁: X₁ {O(n)}
t₃, X₂: X₂ {O(n)}
t₃, X₃: X₃ {O(n)}
t₄, X₀: 2⋅X₁⋅X₂+2⋅X₂⋅X₃+2⋅X₃+6⋅X₂ {O(n^2)}
t₄, X₁: 2⋅X₁ {O(n)}
t₄, X₂: 2⋅X₂ {O(n)}
t₄, X₃: 2⋅X₃ {O(n)}
t₁, X₀: X₃ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₅, X₀: 2⋅X₁⋅X₂+2⋅X₂⋅X₃+6⋅X₂+X₃ {O(n^2)}
t₅, X₁: X₁ {O(n)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: X₃ {O(n)}
t₆, X₀: 2⋅X₁⋅X₂+2⋅X₂⋅X₃+3⋅X₃+6⋅X₂ {O(n^2)}
t₆, X₁: 3⋅X₁ {O(n)}
t₆, X₂: 3⋅X₂ {O(n)}
t₆, X₃: 3⋅X₃ {O(n)}